# Contributions on completely regular semigroups

#### Abstract

Semigroup theory typically looks at algebraic structures which are generalizations of groups. An example of such a generalization is a semigroup which is a union of groups. When the usual semigroup multiplication is supplemented with a unary operation which selects the group inverse of an element, the class of completely regular semigroups forms a variety, or equationally defined class, of unary semigroups. In this dissertation, we look at several problems which concern varieties of completely regular semigroups or closely related objects. The standard examples of varieties of completely regular semigroups are groups, completely simple semigroups, and idempotent semigroups (bands). The lattice of band varieties has been completely described. We make use of the list of inequivalent identities this lattice provides to consider the question of whether and when band free products lie in a proper subvariety of bands. We obtain an example of a band free product of two normal bands which lies in no proper subvariety. We also show that, in some special cases, the band free product of two bands will lie in a proper subvariety. In another problem, we consider a particular multiplication of varieties, known as a Mal'cev product. For arbitrary completely regular varieties, this construction does not in general yield a variety, but it has previously been shown to yield a variety when the first component of a product is a variety of rectangular groups, and the second one, an arbitrary variety. We extend this result to allow the first component to be a variety of central completely simple semigroups. The largest part of the dissertation is devoted to the topic of the lattice of completely regular monoid varieties, denoted ${\cal L}$(MCR). Paralleling a previous result for the lattice of completely regular semigroup varieties ${\cal L}$(CR), we show that ${\cal L}$(MCR) can be represented as a lattice of order-preserving mappings from a certain ordered set into a slightly modified subset of ${\cal L}$(MCR). Our result is somewhat simpler than the original result for ${\cal L}$(CR) because ${\cal L}$(MCR) is isomorphic to a sublattice of ${\cal L}$(CR) which exhibits special properties with respect to the mapping representation, and these carry over to our representation.

#### Recommended Citation

Vachuska, Colleen Ann, "Contributions on completely regular semigroups" (1990). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9117362.

https://epublications.marquette.edu/dissertations/AAI9117362