Pregroup representable expansions of residuated lattices

TL;DR

This paper introduces a method to construct distributive involutive FL-algebras from pregroups, providing relational representations that extend to non-Boolean lattice reducts and distributive quasi relation algebras.

Contribution

It develops a novel construction linking pregroups to DInFL-algebras and demonstrates their relational representability, including for finite pregroups.

Findings

Relational representations of DInFL-algebras from pregroups

Representation of non-Boolean lattice reducts in finite pregroups

Extension to distributive quasi relation algebras with unary operations

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.