Implication Algebras and Implication Semigroups of Binary Relations

TL;DR

This paper explores the properties of implication algebras and semigroups of binary relations, establishing decidability results for implication algebras and demonstrating undecidability for implication semigroups.

Contribution

It proves finite axiomatizability and decidability for representable implication algebras and introduces a Stone-style representation theorem, contrasting with the undecidability results for implication semigroups.

Findings

Finite axiomatizability of representable implication algebras

Decidability of the finite representation problem for implication algebras

Undecidability of the representation decision problem for implication semigroups

Abstract

Representable implication algebras are known to be axiomatised by a finite number of equations (making the representation and finite representation problems decidable here). We show that this also holds in the context of unary (and binary) relations and present a Stone-style representation theorem. We then show that the (finite) representation decision problem is undecidable for implication semigroups, in stark contrast with implication algebras.