Robin Hirsch

A theorem of Monk states that the class of representable relation algebras cannot be defined by a finite set of axioms. We would like to find a subsignature of the signature of relation algebras, so that the corresponding representation class is finitely axiomatisable, while the chosen subsignature is as expressive as possible.

We investigate the signature {., 1', ;} with intersection, identity and composition only. This is the signature of semilattice ordered monoids, sometimes called the "Jerry Fragment" after Jerry Seligman. We show that no finite set of axioms can define the class of representable semilattice ordered monoids.

A related signature is {<=, 1', ;}, the signature of ordered monoids. Whether the class of representable ordered monoids is finitely axiomatisable or not remains open.