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Pregroup representable expansions of residuated lattices

Andrew Craig, Claudette Robinson

TL;DR

The paper advances the representability of algebraic structures beyond relation algebras by proving that distributive involutive FL-algebras ($DInFL$) and distributive quasi relation algebras ($DqRA$) can be realized as algebras of binary relations built from pregroups and their generalizations. Central to the approach are up-set constructions $ extsf{Up}(P, ext{≤})$ and $ extsf{Up}(P^2,owtie)$, together with the $oldsymbol{igtriangleup}$-map (a Cayley-like embedding) and its preservation properties under Conditions such as $W$; for pregroups these embeddings preserve the linear negations, yielding representability for $DInFL$, while ortho pregroups enable $DqRA$ representations via the extended $ eg$-operator. The work provides explicit finite representations from products of groups, illustrating the method and suggesting broader applicability to language-analytic and computational contexts. It also outlines future directions, including partially defined algebraic structures and potential non-representability results, which could further illuminate the boundary between representable and non-representable $DInFL$- and $DqRA$-type algebras.

Abstract

Group representable relation algebras play an important role in the study of representable relation algebras. The class of distributive involutive FL-algebras (DInFL-algebras) generalises relation algebras, as well as Sugihara monoids and MV-algebras. We construct DInFL-algebras from pregroups and show that they can be represented as algebras of binary relations. Even for finite pregroups we obtain relational representations of DInFL-algebras with non-Boolean lattice reducts. If the pregroup is enriched with a particular unary order-reversing operation, then our construction yields representation results for distributive quasi relation algebras.

Pregroup representable expansions of residuated lattices

TL;DR

The paper advances the representability of algebraic structures beyond relation algebras by proving that distributive involutive FL-algebras () and distributive quasi relation algebras () can be realized as algebras of binary relations built from pregroups and their generalizations. Central to the approach are up-set constructions and , together with the -map (a Cayley-like embedding) and its preservation properties under Conditions such as ; for pregroups these embeddings preserve the linear negations, yielding representability for , while ortho pregroups enable representations via the extended -operator. The work provides explicit finite representations from products of groups, illustrating the method and suggesting broader applicability to language-analytic and computational contexts. It also outlines future directions, including partially defined algebraic structures and potential non-representability results, which could further illuminate the boundary between representable and non-representable - and -type algebras.

Abstract

Group representable relation algebras play an important role in the study of representable relation algebras. The class of distributive involutive FL-algebras (DInFL-algebras) generalises relation algebras, as well as Sugihara monoids and MV-algebras. We construct DInFL-algebras from pregroups and show that they can be represented as algebras of binary relations. Even for finite pregroups we obtain relational representations of DInFL-algebras with non-Boolean lattice reducts. If the pregroup is enriched with a particular unary order-reversing operation, then our construction yields representation results for distributive quasi relation algebras.
Paper Structure (11 sections, 24 theorems, 18 equations, 1 figure)

This paper contains 11 sections, 24 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Residuated lattice expansions
  4. Involutive Partially Ordered Monoids
  5. Pregroups
  6. Distributive residuated lattices of binary relations
  7. Representable DInFL-algebras and DqRAs
  8. Representability results for DInFL-algebras
  9. Representations of DqRAs via ortho pregroups
  10. Examples from products of groups
  11. Future work

Key Result

lemma 1

GJL23 Let $\mathbf{P} = \langle P, \leqslant, \cdot, 1, ^-, ^{\sim}\rangle$ be an ipo-monoid. Then the following hold for all $x, y, z\in P$:

Figures (1)

  • Figure 1: The DqRAs $\mathbf{A}_3$ and $\mathbf{A}_4$. They have the identifiers $D^6_{4,8}$ and $D^8_{7,21}$ (see CJR-DqRAs). The black nodes denote idempotent elements.

Theorems & Definitions (43)

  • definition 1
  • lemma 1
  • proposition 1
  • proof
  • definition 2
  • proposition 2
  • proposition 3
  • proof
  • proposition 4
  • lemma 2
  • ...and 33 more