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Realization Boundaries

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255 symbol spellings in 1033 mathematical passages across 24 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.

All 255 entries are available below.

+Notation table · 48 passages

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=Notation table · 152 passages

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ANotation table · 261 passages

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BNotation table · 100 passages

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CNotation table · 65 passages

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cNotation table · 10 passages

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c_ANotation table · 9 passages

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C_pNotation table · 23 passages

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DNotation table · 128 passages

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D_BNotation table · 20 passages

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eNotation table · 35 passages

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e_pNotation table · 10 passages

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\eqfinNotation table · 11 passages

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\FNotation table · 7 passages

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\F_pNotation table · 4 passages

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GNotation table · 119 passages

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\mathfrak GNotation table · 13 passages

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gNotation table · 23 passages

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G_ANotation table · 93 passages

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HNotation table · 55 passages

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H_WNotation table · 27 passages

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KNotation table · 15 passages

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K_{\mathfrak G}Notation table · 6 passages

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mNotation table · 122 passages

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\midNotation table · 23 passages

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\mid_GNotation table · 1 passage

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\NNotation table · 26 passages

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nNotation table · 31 passages

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\oneNotation table · 6 passages

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\one_ANotation table · 3 passages

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PNotation table · 97 passages

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pNotation table · 124 passages

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p_{2n+\one_A(n)}Notation table · 1 passage

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\QNotation table · 34 passages

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qNotation table · 27 passages

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q^{-1}Notation table · 8 passages

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rNotation table · 60 passages

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\radNotation table · 13 passages

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SNotation table · 28 passages

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sNotation table · 37 passages

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S_ANotation table · 20 passages

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s_ANotation table · 32 passages

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TNotation table · 14 passages

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tNotation table · 18 passages

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\triangleNotation table · 2 passages

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uNotation table · 49 passages

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u_ANotation table · 42 passages

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VNotation table · 18 passages

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WNotation table · 47 passages

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\widehat GNotation table · 2 passages

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XNotation table · 25 passages

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yNotation table · 18 passages

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\toNotation table · 25 passages

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\inNotation table · 52 passages

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\prodNotation table · 9 passages

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\prod_pNotation table · 4 passages

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\inftyNotation table · 4 passages

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\leNotation table · 17 passages

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\le_TNotation table · 6 passages

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\subseteqNotation table · 27 passages

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\bigoplusNotation table · 7 passages

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\bigoplus_pNotation table · 3 passages

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dDefinition passage · 37 passages

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 47

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

    Exact mathematical passage
    \lvert d\rvert
\EndDefinition passage · 6 passages

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  • Definition passage: Operator-seed realization · Read passage · 01_contracts.tex, line 11

    [Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In pa…

    Exact mathematical passage
    \mathcal O\subseteq\End(M)
hDefinition passage · 29 passages

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  • Definition passage: Purity · Read passage · 04_purity.tex, line 19

    [Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.

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    my=h
\infDefinition passage · 1 passage

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 48

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

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K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\lvertDefinition passage · 9 passages

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 47

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

    Exact mathematical passage
    \lvert d\rvert
MDefinition passage · 21 passages

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  • Definition passage: Operator-seed realization · Read passage · 01_contracts.tex, line 11

    [Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In pa…

    Exact mathematical passage
    M
\mathcal ODefinition passage · 3 passages

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  • Definition passage: Operator-seed realization · Read passage · 01_contracts.tex, line 11

    [Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In pa…

    Exact mathematical passage
    \mathcal O\subseteq\End(M)
\operatorname{Realize}Definition passage · 1 passage

Search aliases: Realize operatorname

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 48

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

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K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\operatorname{Realize}_{\mathfrak G}Definition passage · 1 passage

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 48

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

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K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\rvertDefinition passage · 9 passages

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 47

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

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    \lvert d\rvert
\trankDefinition passage · 2 passages

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  • Definition passage: Topological generation · Read passage · 01_contracts.tex, line 28

    [Topological generation] For a topological group M, a subset S topologically generates M when the closure of the subgroup generated by S is all of M. Its least possible size is denoted trank(M). An operator version takes the closure of the operator hull instead. The closure is taken in the stated topology. A change of topology is a change of contract, not an innocuous notational adjustment.

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    \trank(M)
\capDefinition passage · 5 passages

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  • Definition passage: Purity · Read passage · 04_purity.tex, line 16

    [Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.

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H\cap mM=mH\qquad(m\ge1).
\congDefinition passage · 20 passages

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  • Definition passage: Grammar-relative description complexity · Read passage · 01_contracts.tex, line 48

    [Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.

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K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\geDefinition passage · 12 passages

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  • Definition passage: Purity · Read passage · 04_purity.tex, line 16

    [Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.

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H\cap mM=mH\qquad(m\ge1).
aIntroduction passage · 7 passages

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
bIntroduction passage · 13 passages

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
b^{-1}Introduction passage · 1 passage

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
B_jIntroduction passage · 8 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 12

    For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.

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    B_j\subseteq\N
d_0Introduction passage · 1 passage

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  • Introduction passage · Read passage · 10_grammar.tex, line 7

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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d_0,d_1,d_2,\ldots
d_1Introduction passage · 1 passage

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  • Introduction passage · Read passage · 10_grammar.tex, line 7

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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d_0,d_1,d_2,\ldots
d_2Introduction passage · 1 passage

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  • Introduction passage · Read passage · 10_grammar.tex, line 7

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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d_0,d_1,d_2,\ldots
G_{B_j}Introduction passage · 1 passage

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  • Introduction passage · Read passage · 10_grammar.tex, line 12

    For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.

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    R_j\cong G_{B_j}
jIntroduction passage · 30 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 10

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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    R_j
\LongleftrightarrowIntroduction passage · 9 passages

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  • Introduction passage · Read passage · 07_classification.tex, line 11

    [Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…

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p\mid_{G_A}g\quad\Longleftrightarrow\quad p\notin S_A.
\mid_{G_A}Introduction passage · 3 passages

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  • Introduction passage · Read passage · 07_classification.tex, line 11

    [Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…

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p\mid_{G_A}g\quad\Longleftrightarrow\quad p\notin S_A.
\nmidIntroduction passage · 3 passages

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
\ObsIntroduction passage · 3 passages

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  • Introduction passage · Read passage · 09_reconstruction.tex, line 6

    The observation contract. Let Obs(G) denote the unmarked inverse system of finite quotient groups and their transition maps, considered up to isomorphism as a pro-object. The source group G is not supplied again as part of the input. Nor do we retain evaluations of named source elements. Supplying the source itself would make a reconstruction problem vacuous, and supplying coherent marked elements is a stronger contract.

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    \Obs(G)
\overline rIntroduction passage · 2 passages

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  • Introduction passage · Read passage · 07_classification.tex, line 92

    The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.

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    \overline r_p
\overline r_pIntroduction passage · 2 passages

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  • Introduction passage · Read passage · 07_classification.tex, line 92

    The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.

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    \overline r_p
\pmodIntroduction passage · 1 passage

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
RIntroduction passage · 6 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 10

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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    R_j
R_jIntroduction passage · 3 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 10

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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    R_j
\srankIntroduction passage · 2 passages

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  • Introduction passage · Read passage · 01_contracts.tex, line 15

    If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.

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    \srank_{\End(M)}(M)
\srank_{\End(M)}Introduction passage · 1 passage

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  • Introduction passage · Read passage · 01_contracts.tex, line 15

    If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.

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    \srank_{\End(M)}(M)
\suppIntroduction passage · 2 passages

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  • Introduction passage · Read passage · 07_classification.tex, line 14

    [Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…

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    \supp(d)
\ldotsIntroduction passage · 5 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 7

    Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.

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d_0,d_1,d_2,\ldots
\varnothingIntroduction passage · 5 passages

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  • Introduction passage · Read passage · 10_grammar.tex, line 12

    For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.

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    B_j=\varnothing
\notinIntroduction passage · 8 passages

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  • Introduction passage · Read passage · 05_construction.tex, line 28

    A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.

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\label{eq:section}
(s_A(a/b))_p=
\begin{cases}
a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
0,&p\notin S_A\text{ or }p\mid b.
\end{cases}
\mathbf0Use in source · 1 passage

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  • Use in source · Read passage · 08_computability.tex, line 81

    ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting degree 0'. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their …

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    \mathbf 0'
\mathbf0'Use in source · 1 passage

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  • Use in source · Read passage · 08_computability.tex, line 81

    ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting degree 0'. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their …

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    \mathbf 0'
A_CUse in source · 9 passages

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  • Use in source · Read passage · 10_grammar.tex, line 17

    Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩),

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    A_C
A_{C'}Use in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 33

    If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.

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    A_{C'}
A_NUse in source · 7 passages

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  • Use in source · Read passage · 02_ambient.tex, line 7

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

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A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
a_nUse in source · 2 passages

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  • Use in source · Read passage · 05_construction.tex, line 6

    Choosing a rational line. Let A⊆N, write a_n=one_A(n), and define S_A={p_2n+a_n:n∈N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,

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    a_n=\one_A(n)
A_{N+1}Use in source · 2 passages

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  • Use in source · Read passage · 02_ambient.tex, line 24

    Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.

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    A_N\to A_{N+1}
\alephUse in source · 4 passages

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  • Use in source · Read passage · 01_contracts.tex, line 37

    [Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a boundary of description, not of existence. It also says nothing about whether a particular named group lies outside a chosen grammar. The later const…

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    \aleph_0
\aleph_0Use in source · 4 passages

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  • Use in source · Read passage · 01_contracts.tex, line 37

    [Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a boundary of description, not of existence. It also says nothing about whether a particular named group lies outside a chosen grammar. The later const…

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    \aleph_0
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  • Use in source · Read passage · 03_spectrum.tex, line 56

    Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…

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    b_0,b_1,\ldots
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  • Use in source · Read passage · 03_spectrum.tex, line 56

    Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…

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    b_0,b_1,\ldots
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  • Use in source · Read passage · 03_spectrum.tex, line 56

    Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…

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    b_j
B_KUse in source · 3 passages

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  • Use in source · Read passage · 03_spectrum.tex, line 66

    Define B_K={2n:n∈N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.

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B_K=\{2n:n\in\N\}\cup\{2n+1:n\notin K\},
\qquad H=D_{B_K}.
\bitsUse in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 31

    Give Sub(D) the topology inherited from {0,1}^D by its membership functions. Membership of a vector with finite support depends on finitely many selector bits. Conversely, membership of e_p_i recovers the ith bit. Consequently the order isomorphism above is also a homeomorphism from Cantor space bits to Sub(D).

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    \bits
\bmodUse in source · 2 passages

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  • Use in source · Read passage · 02_ambient.tex, line 10

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

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    k\bmod m_N
C'Use in source · 2 passages

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  • Use in source · Read passage · 10_grammar.tex, line 33

    If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.

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    C\ne C'
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  • Use in source · Read passage · 05_construction.tex, line 125

    [Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its finite support places it inside a finite cyclic group.

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    C_1
C_{12}Use in source · 1 passage

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  • Use in source · Read passage · 11_atlas.tex, line 71

    For m=12, only coordinates 2 and 3 matter. The six representatives are 0, e_2, e_3, 2e_3, e_2+e_3, e_2+2e_3. The vector e_2+e_3 generates this quotient, which is C_6, not C_12. Repeated prime powers in the multiplier do not create additional coordinates.

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    C_{12}
C_2Use in source · 4 passages

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C_{210}Use in source · 1 passage

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C_{2310}Use in source · 1 passage

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C_3Use in source · 2 passages

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  • Use in source · Read passage · 11_atlas.tex, line 36

    At the unselected prime 3, the coordinate of u_A is zero. In this case 3(1/3,0)=(1,0). The three roots are (1/3,0), (1/3,e_3), (1/3,2e_3). Their differences form G_A[3]=C_3, as predicted by Proposition~[source reference].

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    G_A[3]=C_3
C_{30}Use in source · 3 passages

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C_6Use in source · 4 passages

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C_mUse in source · 2 passages

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  • Use in source · Read passage · 05_construction.tex, line 125

    [Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its finite support places it inside a finite cyclic group.

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    \Z\oplus C_m
C_{p_i}Use in source · 2 passages

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  • Use in source · Read passage · 02_ambient.tex, line 7

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

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A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
C_{p_n}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 84

    A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.

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    C_{p_n}
C_{\rad(m)}Use in source · 4 passages

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  • Use in source · Read passage · 00_notation.tex, line 33

    The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.

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\Tor(G_A)&=D,&G_A/D&\cong\Q,\\
G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\
G_A\cong G_B&\iff A\eqfin B,&&\\
G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
D_{B_K}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 66

    Define B_K={2n:n∈N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.

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B_K=\{2n:n\in\N\}\cup\{2n+1:n\notin K\},
\qquad H=D_{B_K}.
D_{B\cap C}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 25

    e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latti…

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D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C},
\qquad D=D_B\oplus D_{\N\setminus B}.
D_{B\cup C}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 25

    e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latti…

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D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C},
\qquad D=D_B\oplus D_{\N\setminus B}.
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  • Use in source · Read passage · 03_spectrum.tex, line 15

    [Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contain…

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D_B\cong D_C\quad\Longleftrightarrow\quad B=C.
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  • Use in source · Read passage · 11_atlas.tex, line 62

    Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.

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d_m(g)=\sum_{p\mid m}g_p e_p.
D_{\N\setminus B}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 25

    e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latti…

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D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C},
\qquad D=D_B\oplus D_{\N\setminus B}.
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  • Use in source · Read passage · 04_purity.tex, line 32

    bijection between rational subspaces of V and pure subgroups H of P containing D. First let W be rational and suppose h∈ H_W∩ mP. Because W is closed under division by m, choose x∈ H_W with q(x)=q(h)/m. Then d=h-mx belongs to D. At a prime dividing m, the coordinate of h is zero because h∈ mP, and that of mx is also zero. Thus d_p=0 whenever pmid m.

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    d_p=0
\degUse in source · 3 passages

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  • Use in source · Read passage · 08_computability.tex, line 17

    [Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~[source reference] guarantees that every other correction coordinate is zero. Form…

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    \deg_T(A)
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  • Use in source · Read passage · 08_computability.tex, line 17

    [Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~[source reference] guarantees that every other correction coordinate is zero. Form…

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    \deg_T(A)
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  • Use in source · Read passage · 05_construction.tex, line 67

    [A finite support bound] For all r,t∈Q, supp c_A(r,t)⊆ {p:pmidden(r)den(t)}. At a prime not dividing either denominator, all three rationals have well-defined reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On an unselected coordinate, all section values are zero. The reduced denominator of r+t has no prime divisor absent from the two original denominators.

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\supp c_A(r,t)\subseteq
\{p:p\mid\den(r)\den(t)\}.
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  • Use in source · Read passage · 04_purity.tex, line 106

    A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it admits an especially sharp classification and computability theorem.

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    \dim_{\Q}W
\dim_{\Q}Use in source · 1 passage

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  • Use in source · Read passage · 04_purity.tex, line 106

    A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it admits an especially sharp classification and computability theorem.

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    \dim_{\Q}W
EUse in source · 6 passages

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    E
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  • Use in source · Read passage · 07_classification.tex, line 52

    [The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.

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    u_A-u_B=e_2-e_3
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  • Use in source · Read passage · 07_classification.tex, line 52

    [The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.

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    u_A-u_B=e_2-e_3
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  • Use in source · Read passage · 11_atlas.tex, line 96

    A finite tail exception. Consider an infinite-order element x=s_A(5/6)+2e_7. An exception set for its signature is {2,3,5,7}. Outside these primes, the denominator is invertible, the numerator is nonzero, and the torsion correction vanishes. Thus the root pattern of x agrees with that of u_A in every later prime pair.

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x=s_A(5/6)+2e_7.
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  • Use in source · Read passage · 07_classification.tex, line 97

    [Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.

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\End(G_A)\cong
E_A=\{\lambda\in P:\exists r\in\Q,\ 
\lambda_p=\overline r_p\text{ for almost every }p\in S_A\}.
e_{p_i}Use in source · 2 passages

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  • Use in source · Read passage · 03_spectrum.tex, line 22

    Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visib…

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    e_{p_i}
FUse in source · 20 passages

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    F
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  • Use in source · Read passage · 02_ambient.tex, line 95

    The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of D. Taking their …

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    f:D\to F
\F_3Use in source · 1 passage

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  • Use in source · Read passage · 11_atlas.tex, line 58

    A torsion correction that cannot be discarded. Take the purely torsion target e_3 and multiplier 2. The root criterion succeeds because the target has zero 2-coordinate. Its rational part is zero, but the answer is not the zero element: 2(0,2e_3)=(0,e_3). In F_3, multiplication by 2 sends 2 to 1. This is the simplest example showing why the finite correction in the root algorithm is necessary even when the rational coordinate causes no difficulty.

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    \F_3
\F_5Use in source · 1 passage

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  • Use in source · Read passage · 11_atlas.tex, line 20

    bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.

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    \F_5
\F_{p_{b_j}}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 56

    Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…

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    \F_{p_{b_j}}
G_{A_C}Use in source · 3 passages

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  • Use in source · Read passage · 10_grammar.tex, line 28

    [Escape within one profinite-completion class] For every C⊆N, the group G_A_C is not isomorphic to any output of G. Distinct C give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion P. Fix a description index j. If R_j is outside the family, it cannot be isomorphic to G_A_C. Otherwise rule [source reference] makes A_C disagree with B_j at every index 2⟨ j,k⟩. These are infinitely many distinct…

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    G_{A_C}
G_BUse in source · 19 passages

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  • Use in source · README.md, line 17

    G_A≅ G_Biff Atriangle B is finite, G_A has an X-computable copyiff A≤_T X.

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G_A\cong G_B\iff A\triangle B\text{ is finite},\qquad
G_A\text{ has an }X\text{-computable copy}\iff A\le_T X.
G_KUse in source · 1 passage

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  • Use in source · Read passage · 08_computability.tex, line 83

    The group G_K still has a finite mathematical definition. Its noncomputability is not a proof that it cannot be named in a finite semantic language. The theorem concerns effective copies with computable operations and equality.

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    G_K
g_pUse in source · 4 passages

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  • Use in source · Read passage · 06_roots.tex, line 17

    [Root or obstruction] Let g=s_A(r)+d∈ G_A and let m≥1. Then g∈ mG_A Longleftrightarrow g_p=0 for every pmid m. If the condition fails, a prime pmid m with g_p≠0 is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof. Necessity follows because multiplication by m is zero in every coordinate C_p with pmid m.

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g\in mG_A\quad\Longleftrightarrow\quad
 g_p=0\quad\text{for every }p\mid m.
\mathfrak HUse in source · 3 passages

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  • Use in source · Read passage · 10_grammar.tex, line 56

    Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a shortest valid G description of G. Its semantic output remains G, and its length has the stated bound. An obstruction to a small grammar can therefore di…

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    \mathfrak H
h_pUse in source · 3 passages

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  • Use in source · Read passage · 04_purity.tex, line 44

    Root conditions do not depend on the subspace. [Ambient root criterion] For every rational subspace W, every h∈ H_W, and every m≥1, h∈ mH_W Longleftrightarrow h_p=0 for all primes pmid m. Multiplication by m on P is zero at primes dividing m and invertible on every other coordinate. Thus the right-hand condition is equivalent to h∈ mP. Apply purity from Theorem~[source reference]. The condition is independent of W, but the set of admissible h is not. This distinction is the reason identical quot…

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h\in mH_W\quad\Longleftrightarrow\quad
h_p=0\text{ for all primes }p\mid m.
H_VUse in source · 1 passage

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  • Use in source · Read passage · 04_purity.tex, line 38

    D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with D serving as the bottom of…

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    H_V=P
H_{\{0\}}Use in source · 1 passage

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  • Use in source · Read passage · 04_purity.tex, line 38

    Given h∈ H and m≥1, let d∈ D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with…

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    H_{\{0\}}=D
\HomUse in source · 1 passage

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  • Use in source · Read passage · 04_purity.tex, line 75

    [Finite-target equivalence] For every finite group F, restriction along D↪ H_W gives a natural bijection Hom(H_W,F)⟶Hom(D,F). Put m=lvert Frvert. Every h∈ H_W agrees modulo mH_W with an element of D, namely its finite coordinate vector on primes dividing m. Every homomorphism to F kills mH_W. Thus two maps agreeing on D agree everywhere.

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\Hom(H_W,F)\longrightarrow\Hom(D,F).
\HullUse in source · 1 passage

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\Hull_{\mathcal O}Use in source · 1 passage

Search aliases: Hull_ O Hull mathcal

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
IUse in source · 1 passage

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    M=\bigoplus_{i\in I}M_i
iUse in source · 17 passages

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    M=\bigoplus_{i\in I}M_i
\iffUse in source · 3 passages

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  • Use in source · README.md, line 17

    G_A≅ G_Biff Atriangle B is finite, G_A has an X-computable copyiff A≤_T X.

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G_A\cong G_B\iff A\triangle B\text{ is finite},\qquad
G_A\text{ has an }X\text{-computable copy}\iff A\le_T X.
kUse in source · 16 passages

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  • Use in source · Read passage · 02_ambient.tex, line 10

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

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    k\bmod m_N
K_{\mathfrak H}Use in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 57

    Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a shortest valid G description of G. Its semantic output remains G, and its length has the stated bound. An obstruction to a small grammar can therefore di…

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K_{\mathfrak H}(G)\le f(K_{\mathfrak G}(G))
LUse in source · 4 passages

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  • Use in source · Read passage · 05_construction.tex, line 128

    [Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its finite support places it inside a finite cyclic group.

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    L\le G_A
m^{-1}Use in source · 3 passages

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  • Use in source · Read passage · 02_ambient.tex, line 107

    [Unique division in the ambient quotient] Multiplication by every positive integer is bijective on V. Thus V has a unique rational vector-space structure extending its additive group structure. The product P has no nonzero divisible subgroup. Fix m≥1 and x∈ P. Define y_p=m^-1x_p when pnmid m, and y_p=0 when pmid m. Then x-my has finite support. Thus q(x)=m q(y), proving surjectivity on V.

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    y_p=m^{-1}x_p
M_iUse in source · 1 passage

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    M=\bigoplus_{i\in I}M_i
M_jUse in source · 2 passages

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
m_NUse in source · 3 passages

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  • Use in source · Read passage · 02_ambient.tex, line 7

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

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A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
NUse in source · 18 passages

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  • Use in source · Read passage · 01_contracts.tex, line 83

    The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish every element once the object is supplied.

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G\longrightarrow \varprojlim G/N.
\N^2Use in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 17

    Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩),

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    \langle j,k\rangle:\N^2\to\N
\neqfinUse in source · 5 passages

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  • Use in source · Read passage · 07_classification.tex, line 80

    [Torsion-image obstruction] If Aneqfin B, every homomorphism G_A→ G_B has image contained in D. Consequently neither group embeds in the other, and neither is a quotient of the other. A homomorphism preserves torsion and induces a map Q→Q on the quotients, necessarily multiplication by a rational r. If r=0, its image lies in D.

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    A\neqfin B
\one_{A_C}Use in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 18

    Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.

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\one_{A_C}(2\langle j,k\rangle)
 &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\
\one_{A_C}(2\langle j,k\rangle+1)
 &=\one_C(j).\label{eq:diagodd}
\one_{B_j}Use in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 18

    Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.

    Exact mathematical passage
    
\one_{A_C}(2\langle j,k\rangle)
 &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\
\one_{A_C}(2\langle j,k\rangle+1)
 &=\one_C(j).\label{eq:diagodd}
\one_CUse in source · 1 passage

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  • Use in source · Read passage · 10_grammar.tex, line 18

    Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.

    Exact mathematical passage
    
\one_{A_C}(2\langle j,k\rangle)
 &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\
\one_{A_C}(2\langle j,k\rangle+1)
 &=\one_C(j).\label{eq:diagodd}
\one_{S_A}Use in source · 2 passages

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  • Use in source · Read passage · 05_construction.tex, line 99

    [One cocycle value reads one coordinate] For every prime p, c_A(1/p,(p-1)/p)=-one_S_A(p)e_p. The support bound leaves only the coordinate p. At that coordinate both fractional lifts are zero, and the lift of their sum, which is one, is one_S_A(p). Dropping the cocycle is therefore not a harmless simplification. The direct product operation on Q× D removes exactly these defects and gives a different, split extension.

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c_A(1/p,(p-1)/p)=-\one_{S_A}(p)e_p.
\overline MUse in source · 1 passage

Search aliases: overline M overline

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  • Use in source · Read passage · 07_classification.tex, line 56

    All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.

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    \overline M_\lambda
\overline M_\lambdaUse in source · 1 passage

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  • Use in source · Read passage · 07_classification.tex, line 56

    All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.

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    \overline M_\lambda
\overline{\langle u\rangle}Use in source · 1 passage

Search aliases: overline⟨ u⟩ overline langle rangle

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  • Use in source · Read passage · 02_ambient.tex, line 73

    A single topological generator after completion. [Topological cyclicity] Let u=(1,1,1,…)∈ P. Then overline⟨ u⟩=P. In particular, trank(P)=1. A basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer k with the specified residues. The element ku belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice. The generator u is absent from D. The topological…

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\overline{\langle u\rangle}=P.
\mathcal PUse in source · 2 passages

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  • Use in source · Read passage · 03_spectrum.tex, line 14

    [Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contain…

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    \mathcal P(\N)
p_0Use in source · 1 passage

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  • Use in source · Read passage · 00_notation.tex, line 5

    Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning

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p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_1Use in source · 1 passage

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  • Use in source · Read passage · 00_notation.tex, line 5

    Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning

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p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_2Use in source · 1 passage

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  • Use in source · Read passage · 00_notation.tex, line 5

    Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning

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p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_{2n}Use in source · 4 passages

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  • Use in source · Read passage · 06_roots.tex, line 116

    The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the selector written in intrinsic group language.

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 p_{2n}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\in A,\label{eq:evenroot}\\
 p_{2n+1}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\notin A.\label{eq:oddroot}
p_{2n+1}Use in source · 3 passages

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  • Use in source · Read passage · 06_roots.tex, line 116

    The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the selector written in intrinsic group language.

    Exact mathematical passage
    
 p_{2n}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\in A,\label{eq:evenroot}\\
 p_{2n+1}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\notin A.\label{eq:oddroot}
p_{2n+a_n}Use in source · 1 passage

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  • Use in source · Read passage · 05_construction.tex, line 7

    Choosing a rational line. Let A⊆N, write a_n=one_A(n), and define S_A={p_2n+a_n:n∈N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,

    Exact mathematical passage
    
S_A=\{p_{2n+a_n}:n\in\N\}.
p_3Use in source · 1 passage

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  • Use in source · Read passage · 00_notation.tex, line 5

    Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning

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p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_{b_j}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 56

    Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…

    Exact mathematical passage
    \F_{p_{b_j}}
P_iUse in source · 2 passages

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

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    P_i
p_iUse in source · 8 passages

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  • Use in source · Read passage · 02_ambient.tex, line 7

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

    Exact mathematical passage
    
A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
P_jUse in source · 2 passages

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

    Exact mathematical passage
    P_jTP_i
p_nUse in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 84

    A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.

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    C_{p_n}
r'Use in source · 2 passages

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  • Use in source · Read passage · 05_construction.tex, line 53

    normal form] Every element of G_A has a unique expression g=s_A(r)+d, (r,d)∈Q× D. Consequently G_A is countably infinite, with a countable normal-form domain that does not depend on the selector. Existence follows from the preimage definition and the displayed quotient identity. If two forms are equal, applying q gives (r-r')q(u_A)=0. The vector q(u_A) is nonzero in a rational vector space, so r=r'. The finite-support terms are then equal. Both Q and D are countable, while the quotient is the in…

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    (r-r')q(u_A)=0
\operatorname{Reach}Use in source · 1 passage

Search aliases: Reach operatorname

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

    Exact mathematical passage
    
\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\operatorname{Reach}_EUse in source · 1 passage

Search aliases: Reach_E operatorname

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

    Exact mathematical passage
    
\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
S_BUse in source · 3 passages

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  • Use in source · Read passage · 07_classification.tex, line 31

    ghtarrow Aeqfin B. When Aeqfin B, the two groups are equal as subgroups of the fixed product P. Suppose f:G_A→ G_B is an isomorphism. The image of u_A has infinite order. An isomorphism preserves the existence of a pth root for every prime p. Combining the exact signature [source reference] with Lemma~[source reference] for f(u_A) gives S_Aeqfin S_B. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.

    Exact mathematical passage
    S_A\eqfin S_B
s_BUse in source · 5 passages

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  • Use in source · Read passage · 07_classification.tex, line 33

    Conversely, suppose Aeqfin B. For every rational r, the vectors s_A(r) and s_B(r) differ only on the finitely many prime pairs where the selectors differ. Thus their difference belongs to D. The descriptions G_A=s_A(Q)+D and G_B=s_B(Q)+D define the same subset of P. The normal forms need not be identical. Their coordinate change is (r,d)⟼ (r,d+s_A(r)-s_B(r)). It fixes the represented ambient element. The correction is finite exactly because the selectors differ only finitely.

    Exact mathematical passage
    s_B(r)
\srank_{\End(G_A)}Use in source · 1 passage

Search aliases: srank_End(G_A) srank End

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  • Use in source · Read passage · 07_classification.tex, line 123

    Infinite seed rank before completion. [Seed-rank separation] For every A, srank_End(G_A)(G_A)=aleph_0. No finite family topologically generates G_A, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one. Each element s_A(r)+d has only finitely many nonzero coordinates outside S_A. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~[source reference], every endomorphism preserves a zero coordinate as …

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\srank_{\End(G_A)}(G_A)=\aleph_0.
\SubUse in source · 4 passages

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  • Use in source · Read passage · 03_spectrum.tex, line 14

    [Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contain…

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    \Sub(D)
\TorUse in source · 4 passages

Search aliases: Tor

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  • Use in source · Read passage · 00_notation.tex, line 33

    The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.

    Exact mathematical passage
    
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\
G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\
G_A\cong G_B&\iff A\eqfin B,&&\\
G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
u_BUse in source · 4 passages

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  • Use in source · Read passage · 07_classification.tex, line 45

    [Marked classification] An isomorphism (G_A,u_A)≅(G_B,u_B) exists if and only if A=B. A marked isomorphism preserves the exact two-prime root test [source reference]--[source reference] at every index. Therefore every bit agrees. The converse is the identity map.

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    (G_A,u_A)\cong(G_B,u_B)
vUse in source · 3 passages

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  • Use in source · Read passage · 02_ambient.tex, line 109

    For injectivity, suppose mx∈ D. Outside the union of the finite support of mx and the prime divisors of m, multiplication by m is invertible and x_p=0. Hence x∈ D and q(x)=0. Unique division defines (a/b)v as the unique solution w of bw=av. The vector-space laws follow by clearing positive denominators and using injectivity of integer multiplication.

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    (a/b)v
\varinjlimUse in source · 1 passage

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  • Use in source · Read passage · 02_ambient.tex, line 25

    Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.

    Exact mathematical passage
    
D=\varinjlim_N A_N=\bigoplus_p C_p,
\qquad
P=\varprojlim_N A_N=\prod_p C_p.
\varinjlim_NUse in source · 1 passage

Search aliases: varinjlim_N varinjlim

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  • Use in source · Read passage · 02_ambient.tex, line 25

    Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.

    Exact mathematical passage
    
D=\varinjlim_N A_N=\bigoplus_p C_p,
\qquad
P=\varprojlim_N A_N=\prod_p C_p.
\varprojlimUse in source · 3 passages

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  • Use in source · Read passage · 01_contracts.tex, line 83

    The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish every element once the object is supplied.

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G\longrightarrow \varprojlim G/N.
\varprojlim_NUse in source · 2 passages

Search aliases: varprojlim_N varprojlim

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  • Use in source · Read passage · 02_ambient.tex, line 25

    Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.

    Exact mathematical passage
    
D=\varinjlim_N A_N=\bigoplus_p C_p,
\qquad
P=\varprojlim_N A_N=\prod_p C_p.
wUse in source · 2 passages

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  • Use in source · Read passage · 02_ambient.tex, line 109

    For injectivity, suppose mx∈ D. Outside the union of the finite support of mx and the prime divisors of m, multiplication by m is invertible and x_p=0. Hence x∈ D and q(x)=0. Unique division defines (a/b)v as the unique solution w of bw=av. The vector-space laws follow by clearing positive denominators and using injectivity of integer multiplication.

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    w
W_AUse in source · 1 passage

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  • Use in source · Read passage · 05_construction.tex, line 15

    0, · p∉ S_A. Because u_A has infinite support, q(u_A)≠0. We can therefore define a rational line and its preimage by W_A=Q q(u_A), G_A=q^-1(W_A). This definition produces an actual group for every set A. It does not claim an effective method for obtaining the selector bits of an arbitrary A.

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W_A=\Q q(u_A),\qquad G_A=q^{-1}(W_A).
\widehat DUse in source · 2 passages

Search aliases: widehat D widehat

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  • Use in source · Read passage · 02_ambient.tex, line 92

    The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of D. Taking their …

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    \widehat D
\widehat fUse in source · 2 passages

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  • Use in source · Read passage · 09_reconstruction.tex, line 68

    [Reflection within the paired-prime family] For a homomorphism f:G_A→ G_B, the induced completion map widehat f is an isomorphism if and only if f is an isomorphism. One implication follows from functoriality. Conversely, suppose widehat f is an isomorphism. By Theorem~[source reference], its coordinate action is multiplication by a diagonal sequence λ. Each λ_p must be nonzero, so f is injective. If Aneqfin B, Theorem~[source reference] would force its image to be torsion, contradicting injecti…

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    \widehat f
\widehat{G_A}Use in source · 2 passages

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  • Use in source · Read passage · 00_notation.tex, line 33

    The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.

    Exact mathematical passage
    
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\
G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\
G_A\cong G_B&\iff A\eqfin B,&&\\
G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
\widehat{H_W}Use in source · 1 passage

Search aliases: widehatH_W widehat

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  • Use in source · Read passage · 04_purity.tex, line 61

    em[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. The coordinate map onto the displayed finite product is surjective already on D. Its kernel is mH_W by Theorem~[source reference]. The transition ass…

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\widehat{H_W}\cong P.
xUse in source · 39 passages

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  • Use in source · Read passage · 02_ambient.tex, line 45

    Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.

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    mx=0
x_{2n+1}Use in source · 1 passage

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  • Use in source · Read passage · 03_spectrum.tex, line 79

    For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not supplied by this enumeration. The conclusion is not absolute finite indescribability. The displayed definition is finite. What fails is the stronger c…

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    x_{2n+1}=0
x_iUse in source · 2 passages

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  • Use in source · Read passage · 03_spectrum.tex, line 79

    For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not supplied by this enumeration. The conclusion is not absolute finite indescribability. The displayed definition is finite. What fails is the stronger c…

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    x_i
x_pUse in source · 3 passages

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  • Use in source · Read passage · 02_ambient.tex, line 45

    Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.

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    x_p=0
\xrightarrowUse in source · 1 passage

Search aliases: xrightarrow

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  • Use in source · Read passage · 05_construction.tex, line 110

    Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.

    Exact mathematical passage
    
\label{eq:extension}
0\longrightarrow D\longrightarrow G_A\xrightarrow{\rho_A}\Q\longrightarrow0,
\qquad \rho_A(s_A(r)+d)=r.
y_0Use in source · 1 passage

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  • Use in source · Read passage · 06_roots.tex, line 43

    [All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordinate factors lie in D⊆ G_A. The kernel is therefore the displayed finite direct sum, whose cardinal is rad(m). Repeated prime powers do not enlarge the…

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y_0+G_A[m],\qquad
G_A[m]=\bigoplus_{p\mid m}C_p.
y_pUse in source · 2 passages

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  • Use in source · Read passage · 02_ambient.tex, line 107

    [Unique division in the ambient quotient] Multiplication by every positive integer is bijective on V. Thus V has a unique rational vector-space structure extending its additive group structure. The product P has no nonzero divisible subgroup. Fix m≥1 and x∈ P. Define y_p=m^-1x_p when pnmid m, and y_p=0 when pmid m. Then x-my has finite support. Thus q(x)=m q(y), proving surjectivity on V.

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    y_p=m^{-1}x_p
\ZUse in source · 2 passages

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  • Use in source · Read passage · 01_contracts.tex, line 8

    The integer group Z illustrates only the first distinction. One generator and its integer multiples produce infinitely many elements. The groups developed below separate the other distinctions as well. Their purpose is not to make infinite objects mysterious. It is to account for the information that a construction actually consumes.

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    \Z
zUse in source · 2 passages

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  • Use in source · Read passage · 08_computability.tex, line 33

    Decoding a marked copy. [Uniform marked decoding] Given an X-computable copy of G_A together with the name of u_A, one can uniformly compute A in X. For input n, search the domain simultaneously for solutions of p_2ny=u_A, p_2n+1z=u_A. Each candidate test is decidable in X. Exactly one equation has a solution by [source reference]--[source reference]. Dovetailing the searches therefore terminates. A solution to the first equation means n∈ A, and a solution to the second means n∉ A. The procedure…

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p_{2n}y=u_A,\qquad p_{2n+1}z=u_A.
\cdotUse in source · 1 passage

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  • Use in source · Read passage · 11_atlas.tex, line 20

    bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.

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    2\cdot3=1
\timesUse in source · 4 passages

Search aliases: × times

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  • Use in source · Read passage · 05_construction.tex, line 47

    [Unique normal form] Every element of G_A has a unique expression g=s_A(r)+d, (r,d)∈Q× D. Consequently G_A is countably infinite, with a countable normal-form domain that does not depend on the selector. Existence follows from the preimage definition and the displayed quotient identity. If two forms are equal, applying q gives (r-r')q(u_A)=0. The vector q(u_A) is nonzero in a rational vector space, so r=r'. The finite-support terms are then equal. Both Q and D are countable, while the quotient i…

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g=s_A(r)+d,\qquad (r,d)\in\Q\times D.
\lambdaUse in source · 18 passages

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  • Use in source · Read passage · 07_classification.tex, line 56

    All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.

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    \lambda\in P
\lambda_pUse in source · 6 passages

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  • Use in source · Read passage · 07_classification.tex, line 61

    [Complete homomorphism description] Every homomorphism f:G_A→ G_B has the form f(g)_p=λ_p g_p (p prime) for a unique λ∈ P. Conversely, coordinate multiplication by λ defines such a homomorphism exactly when q(λ u_A)∈Q q(u_B). Equivalently, there is a rational r for which λ u_A-s_B(r) has finite support. For a homomorphism f, the composite G_A→ G_B→ C_p kills pG_A. The quotient G_A/pG_A is canonically C_p. Every endomorphism of this cyclic additive group is multiplication by a unique scalar λ_p. …

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f(g)_p=\lambda_p g_p\qquad(p\text{ prime})
\lambda_p^{-1}Use in source · 1 passage

Search aliases: λ_p^-1 lambda

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  • Use in source · Read passage · 07_classification.tex, line 113

    [Automorphisms] An endomorphism with diagonal λ∈ E_A is an automorphism if and only if every coordinate λ_p is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal λ_p^-1. An automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, the eventual rational scalar cannot be zero because S_A is infinite. Outside finitely many selected primes, the inverse diagonal is reduction of its rec…

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    \lambda_p^{-1}
\piUse in source · 1 passage

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  • Use in source · Read passage · 06_roots.tex, line 58

    The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root criterion identifies the kernel. The Chinese remainder theorem identifies the finite product with the squarefree cyclic group. The transition assertion…

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\pi_m:G_A\longrightarrow\prod_{p\mid m}C_p
\pi_mUse in source · 1 passage

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  • Use in source · Read passage · 06_roots.tex, line 58

    The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root criterion identifies the kernel. The Chinese remainder theorem identifies the finite product with the squarefree cyclic group. The transition assertion…

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\pi_m:G_A\longrightarrow\prod_{p\mid m}C_p
\rhoUse in source · 2 passages

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  • Use in source · Read passage · 05_construction.tex, line 110

    Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.

    Exact mathematical passage
    
\label{eq:extension}
0\longrightarrow D\longrightarrow G_A\xrightarrow{\rho_A}\Q\longrightarrow0,
\qquad \rho_A(s_A(r)+d)=r.
\rho_AUse in source · 2 passages

Search aliases: ρ_A rho

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  • Use in source · Read passage · 05_construction.tex, line 110

    Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.

    Exact mathematical passage
    
\label{eq:extension}
0\longrightarrow D\longrightarrow G_A\xrightarrow{\rho_A}\Q\longrightarrow0,
\qquad \rho_A(s_A(r)+d)=r.
\sigmaUse in source · 2 passages

Search aliases: σ sigma

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  • Use in source · Read passage · 03_spectrum.tex, line 10

    The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.

    Exact mathematical passage
    \sigma_p(M)=1
\sigma_pUse in source · 1 passage

Search aliases: σ_p sigma

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  • Use in source · Read passage · 03_spectrum.tex, line 10

    The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.

    Exact mathematical passage
    \sigma_p(M)=1
\sigma_{p_i}Use in source · 1 passage

Search aliases: σ_p_i sigma

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  • Use in source · Read passage · 03_spectrum.tex, line 22

    Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visib…

    Exact mathematical passage
    \sigma_{p_i}(D_B)=1
\mapstoUse in source · 4 passages

Search aliases: ↦ mapsto

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  • Use in source · Read passage · 03_spectrum.tex, line 14

    [Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contain…

    Exact mathematical passage
    B\mapsto D_B
\hookrightarrowUse in source · 4 passages

Search aliases: ↪ hookrightarrow

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  • Use in source · Read passage · 04_purity.tex, line 68

    If H_W→ F is a homomorphism to a finite group, then the image is abelian and is killed by lvert Frvert. Its kernel therefore contains lvert Frvert H_W. Hence the displayed kernels are cofinal. Their inverse limit is the product of all prime coordinates, which is P. Under this identification the completion map is the original inclusion H_W↪ P. There is a stronger natural statement. It compares maps to every finite target rather than merely comparing the quotient groups.

    Exact mathematical passage
    H_W\hookrightarrow P
\existsUse in source · 1 passage

Search aliases: ∃ exists

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  • Use in source · Read passage · 07_classification.tex, line 97

    [Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.

    Exact mathematical passage
    
\End(G_A)\cong
E_A=\{\lambda\in P:\exists r\in\Q,\ 
\lambda_p=\overline r_p\text{ for almost every }p\in S_A\}.
\prod_{i<N}Use in source · 1 passage

Search aliases: ∏_i<N prod

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  • Use in source · Read passage · 02_ambient.tex, line 7

    Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.

    Exact mathematical passage
    
A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
\prod_{p\mid m}Use in source · 4 passages

Search aliases: ∏_pmid m prod mid

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  • Use in source · Read passage · 04_purity.tex, line 57

    [One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. The coordinate map onto the displayed finite product is surjective already on D. Its kernel is mH_W by Theorem~[source reference]. The transition asser…

    Exact mathematical passage
    
H_W/mH_W\ \cong\ \prod_{p\mid m}C_p.
\sumUse in source · 1 passage

Search aliases: ∑ sum

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  • Use in source · Read passage · 11_atlas.tex, line 62

    Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.

    Exact mathematical passage
    
d_m(g)=\sum_{p\mid m}g_p e_p.
\sum_{p\mid m}Use in source · 1 passage

Search aliases: ∑_pmid m sum mid

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  • Use in source · Read passage · 11_atlas.tex, line 62

    Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.

    Exact mathematical passage
    
d_m(g)=\sum_{p\mid m}g_p e_p.
\setminusUse in source · 3 passages

Search aliases: ∖ setminus

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  • Use in source · Read passage · 02_ambient.tex, line 45

    Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.

    Exact mathematical passage
    P\setminus D
\cupUse in source · 2 passages

Search aliases: ∪ cup

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  • Use in source · Read passage · 03_spectrum.tex, line 25

    e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latti…

    Exact mathematical passage
    
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C},
\qquad D=D_B\oplus D_{\N\setminus B}.
\neUse in source · 9 passages

Search aliases: ≠ ne

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  • Use in source · Read passage · 01_contracts.tex, line 70

    ⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive operator can move its support only al…

    Exact mathematical passage
    M_j\ne0
\oplusUse in source · 3 passages

Search aliases: ⊕ oplus

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  • Use in source · Read passage · 03_spectrum.tex, line 25

    e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latti…

    Exact mathematical passage
    
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C},
\qquad D=D_B\oplus D_{\N\setminus B}.
\bigoplus_{i\in B}Use in source · 1 passage

Search aliases: ⊕_i∈ B bigoplus in

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  • Use in source · Read passage · 03_spectrum.tex, line 7

    The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.

    Exact mathematical passage
    
D_B=\bigoplus_{i\in B}C_{p_i}\le D.
\bigoplus_{i\in I}Use in source · 1 passage

Search aliases: ⊕_i∈ I bigoplus in

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  • Use in source · Read passage · 01_contracts.tex, line 65

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

    Exact mathematical passage
    M=\bigoplus_{i\in I}M_i
\bigoplus_{j\in\operatorname{Reach}_E(F)}Use in source · 1 passage

Search aliases: ⊕_j∈Reach_E(F) bigoplus in operatorname

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  • Use in source · Read passage · 01_contracts.tex, line 66

    [Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…

    Exact mathematical passage
    
\Hull_{\mathcal O}(S)\subseteq
\bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\bigoplus_{p\mid m}Use in source · 1 passage

Search aliases: ⊕_pmid m bigoplus mid

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  • Use in source · Read passage · 06_roots.tex, line 43

    [All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordinate factors lie in D⊆ G_A. The kernel is therefore the displayed finite direct sum, whose cardinal is rad(m). Repeated prime powers do not enlarge the…

    Exact mathematical passage
    
y_0+G_A[m],\qquad
G_A[m]=\bigoplus_{p\mid m}C_p.
\langleUse in source · 4 passages

Search aliases: ⟨ langle

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  • Use in source · Read passage · 05_construction.tex, line 130

    Otherwise choose a lift x∈ L of a generator of its rational image. Subtracting suitable integer multiples of x from the given generators produces finitely many elements of D. They generate the kernel of L→ρ_A(L), which is finite cyclic of squarefree order. The subgroup ⟨ x⟩ intersects that torsion kernel trivially, giving the direct sum. The extensions over finitely generated rational images split, but the extension over all of Q does not. This is another exact distinction between finite algebra…

    Exact mathematical passage
    \langle x\rangle
\rangleUse in source · 4 passages

Search aliases: ⟩ rangle

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  • Use in source · Read passage · 05_construction.tex, line 130

    Otherwise choose a lift x∈ L of a generator of its rational image. Subtracting suitable integer multiples of x from the given generators produces finitely many elements of D. They generate the kernel of L→ρ_A(L), which is finite cyclic of squarefree order. The subgroup ⟨ x⟩ intersects that torsion kernel trivially, giving the direct sum. The extensions over finitely generated rational images split, but the extension over all of Q does not. This is another exact distinction between finite algebra…

    Exact mathematical passage
    \langle x\rangle
\longrightarrowUse in source · 5 passages

Search aliases: ⟶ longrightarrow

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  • Use in source · Read passage · 01_contracts.tex, line 83

    The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish every element once the object is supplied.

    Exact mathematical passage
    
G\longrightarrow \varprojlim G/N.
\longmapstoUse in source · 2 passages

Search aliases: ⟼ longmapsto

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  • Use in source · Read passage · 04_purity.tex, line 26

    [Pure-preimage correspondence] The map W⟼ H_W=q^-1(W) is an inclusion-preserving bijection between rational subspaces of V and pure subgroups H of P containing D. First let W be rational and suppose h∈ H_W∩ mP. Because W is closed under division by m, choose x∈ H_W with q(x)=q(h)/m. Then d=h-mx belongs to D. At a prime dividing m, the coordinate of h is zero because h∈ mP, and that of mx is also zero. Thus d_p=0 whenever pmid m.

    Exact mathematical passage
    
W\longmapsto H_W=q^{-1}(W)

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