Peter Higgins

Algebras defined by equations

In a recent paper with Marcel Jackson we proved that a class of algebras closed under the taking of homomorphisms, direct products, and elementary embeddings can be defined by a list of equations of a type that will be explained.

In this talk we concentrate on how this applies to classes of semigroup, which was the original motivation for the theorem, as many intensively studied semigroup classes are EHP-classes, as we call them, without being varieties, which is to say the class in question is not closed under the taking of arbitrary subsemigroups.

EHP classes include Regular Semigroups, Groups, Left and Right groups, Unions of groups, Cryptogroups, Semilattices of groups, Completely simple semigroups, Monoids, Left-, Right-, and J-Simple semigroups (but not Bisimple semigroups), Inverse, Orthodox, Conventional, and E-solid semigroups. In some cases equational bases are obvious, but not so in others. We examine the link between e-varieties in the sense of Hall and EHP classes consisting of regular semigroups.

There is scope for exploiting automated theorem provers as we are working in first order languages. Indeed we give an example of an EHP basis for completely regular semigroups that was first verified by AI.

Finally we shall introduce a dual to the Birkhoff theorem on varieties as applied to EHP-classes of semigroups closed under the taking of containing semigroups.