Formal languages and idempotent semigroups

Abstract

The structure of the lattice 𝗟𝗕 of varieties of idempotent semigroups or bands (as universal algebras) was determined by Birjukov, Fennemore and Gerhard. Wis- math determined the structure of a related lattice: the lattice LBM of varieties of band monoids. In the first two parts we study several questions about these varieties.

In Part I we compute the cardinalities of the Green classes of the free objects in each variety of 𝗟𝗕 [𝗟𝗕𝗠]. These cardinalities constitute a useful piece of information in the study of several questions about these varieties and some of the conclusions obtained here are used in parts II and III.

Part II concerns expansions of bands [band monoids]. More precisely, we compute here the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕. We define Rhodes expansion of a monoid, its cut-down to generators and we compute the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕𝗠.

In Part III we deal with Eilenberg varieties of band monoids. The last chapter is particularly concerned with the description of the varieties of languages corresponding to these varieties.