Implication Algebras and Implication Semigroups of Binary Relations

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.