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Generic groups and the weak amalgamation property

Aleksander Ivanov, Krzysztof Majcher

TL;DR

The paper develops a universal, model-theoretic framework based on weak amalgamation (WAP) and joint embedding (JEP) to study generic objects in the logic spaces of countable enumerated groups, semigroups, and rings. It provides general meta-theorems showing when generics cannot exist (via undecidability and Kuznetsov-type arguments) and proves positive results identifying generic groups in abelian and certain nilpotent exponent-p varieties. It also demonstrates nonexistence results for generic group actions on (Q, <) and extends the approach to semigroups and rings, where no generic structures are found. Overall, the work unifies and extends prior results on generic objects, offering a versatile toolkit for analyzing genericity across algebraic structures.

Abstract

We consider the logic space of countable (enumerated) groups and show that closed subspaces corresponding to some standard classes of groups have (do not have) generic groups. We also discuss the cases of semigroups and associative rings.

Generic groups and the weak amalgamation property

TL;DR

The paper develops a universal, model-theoretic framework based on weak amalgamation (WAP) and joint embedding (JEP) to study generic objects in the logic spaces of countable enumerated groups, semigroups, and rings. It provides general meta-theorems showing when generics cannot exist (via undecidability and Kuznetsov-type arguments) and proves positive results identifying generic groups in abelian and certain nilpotent exponent-p varieties. It also demonstrates nonexistence results for generic group actions on (Q, <) and extends the approach to semigroups and rings, where no generic structures are found. Overall, the work unifies and extends prior results on generic objects, offering a versatile toolkit for analyzing genericity across algebraic structures.

Abstract

We consider the logic space of countable (enumerated) groups and show that closed subspaces corresponding to some standard classes of groups have (do not have) generic groups. We also discuss the cases of semigroups and associative rings.
Paper Structure (14 sections, 15 theorems, 8 equations)

This paper contains 14 sections, 15 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. General preliminaries
  3. The case of groups
  4. I. The basic case
  5. II. The case of an action
  6. Generics are generic
  7. Seeking generics
  8. t-Isolation
  9. Abelian groups
  10. Nilpotent groups of prime odd exponent
  11. Strong undecidability implies non-WAP
  12. Completeness and Kuznetsov's theorem
  13. Generics over rationals
  14. Semigroups and rings

Key Result

Proposition 1.3

The set of $U$-generic groups is an invariant comeagre subset of $\mathcal{G}_U$. The same property holds for the set of existentially closed groups.

Theorems & Definitions (26)

  • Remark 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 16 more