Representable and diagonally representable weakening relation algebras

TL;DR

This paper studies weakening relation algebras on posets, introduces a two-player game for their representability, and characterizes classes of these algebras, including diagonally representable ones, with explicit axiomatizations and representations.

Contribution

It introduces a game-theoretic approach to representability of weakening relation algebras and characterizes diagonally representable algebras as a discriminator variety.

Findings

A two-player game characterizes representability.

Explicit finite axiomatizations for certain classes.

Diagonally representable weakening relation algebras form a discriminator variety.

Abstract

A binary relation defined on a poset is a weakening relation if the partial order acts as a both-sided compositional identity. This is motivated by the weakening rule in sequent calculi and closely related to models of relevance logic. For a fixed poset the collection of weakening relations is a subreduct of the full relation algebra on the underlying set of the poset. We present a two-player game for the class of representable weakening relation algebras akin to that for the class of representable relation algebras. This enables us to define classes of abstract weakening relation algebras that approximate the quasivariety of representable weakening relation algebras. We give explicit finite axiomatisations for some of these classes. We define the class of diagonally representable weakening relation algebras and prove that it is a discriminator variety. We also provide explicit representations for several small weakening relation algebras.