Glossary

Admissible realization

The particular object or image allowed by a construction contract, as distinct from every compatible tuple in an ambient completion.

Atomic diagram

The effective record of the basic relations and operations of a countable named structure. It is the standard object used to measure the degree of a copy.

Cocycle

Here, a symmetric map \(c:{\mathbb Q}\times{\mathbb Q}\to D\) satisfying the identity that makes the twisted addition on \({\mathbb Q}\times D\) associative.

Cofinal quotient system

A family of finite quotients through which every finite quotient factors. Its inverse limit computes the full profinite completion.

Computable copy

A countable realization with effective names, decidable equality, and computable operations. An oracle version permits the stated oracle.

Computable profinite presentation

An effective inverse system of finite groups and surjective transition maps whose limit is the given profinite group. This is not a computable enumeration of all elements of an uncountable product.

Computably enumerable set

A set whose members can be listed by a procedure. Nonmembership need not be decidable.

Completion

An ambient object obtained by admitting compatible limits in a specified topology. In this book the relevant completion is profinite.

Dense subgroup

A subgroup meeting every nonempty basic open set. Density does not imply equality with the ambient group.

Divisible abelian group

An abelian group in which every equation \(my=x\) with \(m\ge1\) is solvable. A divisible quotient need not split off as a subgroup.

Endomorphism-seed rank

The minimum number of elements generating a group as a module over its full endomorphism ring.

Eventual equality

Equality modulo finitely many coordinates, written \(A{=^{*}} B\). This is the unmarked selector invariant of \(G_A\).

Finite description

A valid finite code in a specified language with declared semantics. It need not provide computable operations.

Finite support

A vector has only finitely many nonzero coordinates. Such vectors form \(D\) inside \(P\).

Fully invariant subgroup

A subgroup preserved by every endomorphism of its ambient group. Every subgroup of the squarefree direct sum has this property.

Grammar-relative nonrealizability

Failure to have an admissible description in one fixed grammar. It is not absolute mathematical nonexistence or indescribability.

Marked group

A group together with named elements that isomorphisms must preserve. The mark \(u_A\) retains the literal selector.

Normal form

The unique pair \((r,d)\) representing \(s_A(r)+d\). Equality of these names is simpler than the selector-dependent addition law.

Oracle

An external set-membership resource available to a computation. Its information is charged to the effective realization contract.

Prime spectrum

The prime orders of nonzero torsion elements. It classifies the groups \(D_B\) but is constant across all \(G_A\).

Profinite completion

The inverse limit of all finite quotient groups, denoted \(\widehat G\). It is not the Pontryagin dual.

Procyclic group

A topological group with a dense cyclic subgroup. The product \(P\) is procyclic.

Pure subgroup

A subgroup \(H\le M\) for which \(H\cap mM=mH\) for all positive integers \(m\).

Rational line

A one-dimensional subspace over \({\mathbb Q}\). The preimage of \({\mathbb Q} q(u_A)\) is \(G_A\).

Reduced group

A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.

Residual finiteness

Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.

Root certificate

Either an exact element \(y\) with \(my=g\), or a prime divisor of \(m\) at which the target has a nonzero coordinate.

Selector

A subset \(A\subseteq{\mathbb N}\) whose bits choose one prime from each consecutive pair. It determines the admissible rational line.

Squarefree radical

The integer \({\operatorname{rad}}(m)\) formed by retaining each prime divisor of \(m\) once.

Tail signature

The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of \(G_A\) has the selector's tail signature.

Torsion

Elements of finite order. In the product and in every \(G_A\), the torsion subgroup is exactly \(D\).

Turing degree

A measure of relative computability under mutual oracle computation. The least presentation degree of \(G_A\) is the degree of \(A\).

Unmarked observations

Finite quotient objects and transition maps without a named coherent source element or the original admissible image.

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