Varieties of completely semisimple inverse semigroups

Abstract

In [10] Vagner essentially described the class of inverse semigroups as a variety. Research into this class as a variety accelerated with the derivation of good descriptions of the free inverse semigroup of rank one by Gluskin [3] and by Eberhart and Selden [2] and of the free inverse semigroup of arbitrary rank by Scheiblich [9]. However, the first substantial insight into the structure of the lattice of varieties was provided by Djadchenko [1] and Kleiman [4], [5]. In particular, they showed that the lattice has three isomorphic layers at the bottom. The first consisting of varieties of groups, the second of varieties of Clifford semigroups and the third of varieties generated by Brandt semigroups. In [5], Kleiman provides a basis of identities for each of these varieties. Whereas there is a natural description of the semigroups in the first two layers referred to above, there is no such description of those appearing in the third layer. The second section of this paper is devoted to obtaining structural information about these semigroups. It is shown, for instance, that any semigroup appearing in a variety contained in f~ v ~P(B~), where f~ is the variety of groups and ~P(B2) is the variety generated by the Brandt semigroup B 2 = J/d°(1; I ; I; A) with [I1 = 2, is a subdirect product of Brandt semigroups, is completely semisimple and satisfies ~-majorization in the sense of Petrich [7]. In Section 3, four different characterizations are given of varieties of inverse semigroups every member of which is completely semisimple and in which is a congruence. Although this latter class, W~c~o~, is not itself a variety, the varieties contained in rg~9°~ form a sublattice of the lattice of all varieties and some aspects of this sublattice are considered. In particular, it is shown that the even smaller sublattice consisting of varieties contained in 7(B21) v is non-modular.