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Nonrepresentable relation algebras from group systems

H. Andréka, S. Givant, I. Németi

TL;DR

The paper constructs a family of nonrepresentable locally functional polygroupoids (LPGs) from systems of groups using group coset frames and a shifting mechanism on coset data. This yields a proof that there are continuum many subvarieties between representable relation algebras ($ ext{RRA}$) and coset relation algebras ($ ext{CRA}$). The core method generalizes Cayley representations to polygroupoids and leverages scaffold-based criteria to certify nonrepresentability. The results illuminate the rich lattice of cosine-structured relation algebras and demonstrate the diversity of coset relation algebras beyond the representable case, with implications for the equational theory and axiomatizability of these classes.

Abstract

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

Nonrepresentable relation algebras from group systems

TL;DR

The paper constructs a family of nonrepresentable locally functional polygroupoids (LPGs) from systems of groups using group coset frames and a shifting mechanism on coset data. This yields a proof that there are continuum many subvarieties between representable relation algebras () and coset relation algebras (). The core method generalizes Cayley representations to polygroupoids and leverages scaffold-based criteria to certify nonrepresentability. The results illuminate the rich lattice of cosine-structured relation algebras and demonstrate the diversity of coset relation algebras beyond the representable case, with implications for the equational theory and axiomatizability of these classes.

Abstract

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

Paper Structure

This paper contains 7 sections, 9 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Groups, groupoids, group systems
  3. Construction of group frames
  4. The normal subgroups
  5. The system of quotient isomorphisms
  6. Shifting, proof of nonrepresentability
  7. Continuum many subvarieties of coset relation algebras

Key Result

Theorem 2.1

$\hbox{\sf LPG}$ is exactly the class of structures associated to group coset frames, up to isomorphisms. In more detail:

Figures (3)

  • Figure 1: Group coset frame $(\langle\hbox{$\mathfrak G$}_x : x\in I\rangle,\langle\varphi_{xy} :(x,y)\in\mathcal{E}\rangle,\langle C_{xyz} : (x,y),(y,z)\in\mathcal{E}\rangle)$.
  • Figure 2: The group frame $(\hbox{$\mathcal{G}$},\varphi)$.
  • Figure 3: Selection of the points in the representation of $\hbox{$\mathfrak M$}$.

Theorems & Definitions (15)

  • Definition 2.1: Polygroupoid, comer1
  • Definition 2.2: Group coset system, ga02, andgiv1
  • Definition 2.3: Group coset frame, andgiv1
  • Definition 2.4: Structure associated to group coset system
  • Theorem 2.1: Group coset frame theorem, andgiv1gaapal
  • Definition 2.5: Representation of polygroupoid
  • Definition 2.6: Scaffold, ga02
  • Theorem 2.2: Group frame theorem, giv1gaapal
  • Lemma 3.1
  • Lemma 3.2
  • ...and 5 more