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Contractions of quasi relation algebras and applications to representability

Andrew Craig, Wilmari Morton, Claudette Robinson

TL;DR

This work extends the theory of relation algebras to quasi relation algebras by introducing contractions $p\mathbf{A}p$ built from positive symmetric idempotents $p$. It shows that contractions of distributive qRAs preserve representability, providing a mechanism to generate new representable algebras from existing ones through a relabeled quotient on a representing poset. The authors exploit Up-structures, equivalence relations, and automorphisms to embed contractions into relational representations, ensuring preservation of all operations, including linear negations. Additionally, they identify criteria for (non-)finite representability, proving that certain DqRAs cannot be represented on finite posets and establishing a contrapositive relationship between the finite representability of $\mathbf{A}$ and its contraction. These results yield a practical method to construct families of representable DqRAs and to recognize non-finitely representable ones, with concrete examples such as $D^6_{3,5,2}$ and related contractions.

Abstract

Quasi relation algebras (qRAs) were first described by Galatos and Jipsen in 2013. They are generalisations of relation algebras and can also be viewed as certain residuated lattice expansions. We identify positive symmetric idempotent elements in qRAs and show that they can be used to construct new qRAs, so-called contractions of the original algebra. We then show that the contraction of a distributive qRA will be representable when the original algebra is representable. Further, we identify a class of distributive qRAs that are not finitely representable.

Contractions of quasi relation algebras and applications to representability

TL;DR

This work extends the theory of relation algebras to quasi relation algebras by introducing contractions built from positive symmetric idempotents . It shows that contractions of distributive qRAs preserve representability, providing a mechanism to generate new representable algebras from existing ones through a relabeled quotient on a representing poset. The authors exploit Up-structures, equivalence relations, and automorphisms to embed contractions into relational representations, ensuring preservation of all operations, including linear negations. Additionally, they identify criteria for (non-)finite representability, proving that certain DqRAs cannot be represented on finite posets and establishing a contrapositive relationship between the finite representability of and its contraction. These results yield a practical method to construct families of representable DqRAs and to recognize non-finitely representable ones, with concrete examples such as and related contractions.

Abstract

Quasi relation algebras (qRAs) were first described by Galatos and Jipsen in 2013. They are generalisations of relation algebras and can also be viewed as certain residuated lattice expansions. We identify positive symmetric idempotent elements in qRAs and show that they can be used to construct new qRAs, so-called contractions of the original algebra. We then show that the contraction of a distributive qRA will be representable when the original algebra is representable. Further, we identify a class of distributive qRAs that are not finitely representable.
Paper Structure (7 sections, 17 theorems, 19 equations, 5 figures, 1 table)

This paper contains 7 sections, 17 theorems, 19 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi relation algebras
  4. Representable DqRAs
  5. Contractions of quasi relation algebras
  6. Representability of contractions of DqRAs
  7. DqRAs that are not finitely representable

Key Result

lemma 1

RDqRA25 Let $E$ be an equivalence relation on a set $X$, and let $R, S, \gamma \subseteq E$. If $\gamma$ satisfies $\gamma^{\smile}\mathbin{;} \gamma = \mathrm{id}_X$ and $\gamma \mathbin{;} \gamma^{\smile}=\mathrm{id}_X$ then the following hold:

Figures (5)

  • Figure 1: $D^6_{3,5,2}$ is a DqRA with positive symmetric idempotents $\top,1,a$ and $b$.
  • Figure 2: The poset used to represent $D^6_{3,5,2}$.
  • Figure 3: The algebras $p\mathbf{A}p$ for $\mathbf{A}=D^6_{3,5,2}$ and $p\in\{1,\top,a,b\}$.
  • Figure 4: The posets $\mathbf X/{\equiv}_p$ for $p\in\{1,a,b,\top\}$
  • Figure 5: DqRAs that are not finitely representable due to Theorem \ref{['Theorem:NoFiniteRep']}.

Theorems & Definitions (30)

  • lemma 1
  • lemma 2
  • proof
  • theorem 1
  • definition 1
  • lemma 3
  • proof
  • lemma 4
  • proof
  • lemma 5
  • ...and 20 more