Two operators on the lattice of completely regular semigroup varieties

Abstract

In this paper, varieties of completely regular semigroups are studied. This paper is divided into six sections. Section 1 contains an introduction to varieties of completely regular semigroups and preliminaries. Most of the notation needed in this paper is given. In Section 2, the operators \(\overleftarrow {\left( {} \right)} \) and \(\overrightarrow {\left( {} \right)} \) on the lattice of subvarieties of varieties of completely regular semigroups are investigated. In Section 3, some further properties of the operators \(\overleftarrow {\left( {} \right)} \) and \(\overrightarrow {\left( {} \right)} \) are given. In Section 4, the semigroups generated by various subset of some operators are considered. In Section 5, the operators \(\overleftarrow {\left( {} \right)} \) and \(\overrightarrow {\left( {} \right)} \) are used in finding the join of two given varieties. The word problem for free objects in the variety OLBG is considered in Section 6 using the operator \(\overleftarrow {\left( {} \right)} \).