The Class of Representable Ordered Monoids has a Recursively Enumerable, Universal Axiomatisation but it is Not Finitely Axiomatisable

An ordered monoid is a structure with an identity element (1′), a binary composition operator (;) and an antisymmetric partial order (≤), satisfying certain axioms. A representation of an ordered monoid is a 1-1 map which maps elements of an ordered monoid to binary relations in such a way that 1′ is mapped to the identity relation, ; corresponds to composition of binary relations and ≤ corresponds to inclusion of binary relations. We devize a two player game that tests the representability of an ordered monoid n times and show that these games characterise representability. From this we obtain a recursively enumerable, universal axiomatisation of the class of all representable ordered monoids. For each n < ω we construct an unrepresentable ordered monoid An and show that the second player has a winning strategy in a game of length n. Hence we prove that the class of all representable ordered monoids is not finitely axiomatisable. Relation Algebras are badly behaved in a number of ways. The class of representable relation algebras cannot be defined by finitely many axioms [Mon64], nor by any set of equations using a finite number of variables [Jón91], nor by any Sahlqvist theory [Ven97], the equational theory of relation algebras and the equational theory of representable relation algebras is undecidable [Tar41], the problem of determining whether a finite relation algebra is representable or not is itself undecidable [HH01]. An important line of research is to consider reducts of relation algebras, by dropping some of the operators from the signature. We aim to find out exactly what causes this “bad behaviour” and how it can be avoided. Mikulás has surveyed much of this research [Mik03]. In the current paper we consider algebras in the reduced signature {≤, 1′, ; }. Such an algebra is representable if its elements can be interpretted as binary relations over some domain in such a way that ≤ is represented as inclusion of