Local Varieties of Completely Regular Monoids
Journal of Algebra
The aim of this paper is to apply recent deep results on completely regular semigroups to questions about categories, arising in connection with the theory of recognizable languages. A variety [pseudovariety] V of monoids is local if any [finite] category whose local monoids belong to V divides a member of V, in the sense of B. Tilson. Our "Main Theorem" (7.1) essentially states that a monoid variety which satisfies an identity xη+l= x, n ⩾0, that is, which consists of completely regular semigroups, is local whenever all the "labels" on its "Polák ladder" are local. (See below for the definitions.) As an immediate consequence, every nontrivial monoid variety consisting of orthogroups is local (An orthogroup is a completely regular semigroup whose idempotents form a subsemigroup.) Further, it quickly follows that any nontrivial pseudovariety of monoids consisting of orthogroups is also local. At the core of the paper is a study of congruences on locally completely regular categories (those for which each local monoid is completely regular).
Pseudovarieties of monoids arise naturally via the well-known correspondence between regular languages and finite monoids. It has lately been realized that, for some purposes, they are best studied within the framework of pseudovarieties of categories; locality of a (pseudo-) variety V of monoids implies that there is a unique (pseudo-) variety of categories whose monoids comprise V. This enables the solution of various decision problems on varieties and pseudovarieties. For example, we deduce that for any nontrivial pseudovariety V of monoids consisting of orthogroups, V * D =LV, in the notation of.