Subsemigroups of completely simple semigroups
Abstract
The class ${\cal CS}\sb{s}$ of all semigroups that are embeddable in completely simple semi-groups forms a quasivariety. That is, it is the class of all semigroups that satisfy some set of implications. One of the problems considered in this dissertation involves establishing a set of implications which defines the quasivariety ${\cal CS}\sb{s}$, and determining whether or not ${\cal CS}\sb{s}$ can be characterized by a finite set of implications. The development of the free completely simple semigroup on a given semigroup plays the vital part in solving this problem. The free completely simple semigroup $(C,\gamma)$ on a given semigroup which satisfies certain implications $(Q\sb{n}),\ n\ge 1$ is constructed, and a set of implications which defines ${\cal CS}\sb{s}$ is obtained by an extensive study of this free completely simple semigroup $(C,\gamma).$ One of the major results of Malcev concerning group embeddable semigroups is employed in solving the other part of the problem. It is proved that no finite set of implications will define ${\cal CS}\sb{s}$. The model of the free completely simple semigroup $(C,\gamma)$ on a given semigroup S is also used to determine the embeddability of S into a completely simple semigroup of right quotients of S. If ${\cal V}$ is a proper subvariety of the variety of all semigroups then the class of all semigroups of ${\cal V}$ which are embeddable in completely simple semigroups forms a quasivariety. As for ${\cal CS}\sb{s}$, similar questions concerning ${\cal V}\cap{\cal CS}\sb{s}$ are treated and a finite set of implications which defines the members of ${\cal V}\cap{\cal CS}\sb{s}$ is presented. For $n\ge 1,$ the class $({\cal CS}({\cal N}\sb{n}))\sb{s}$ of all semigroups that are embeddable in completely simple semigroups over nilpotent groups of class n forms a quasivariety. The approach of Neumann and Taylor regarding semigroups that are embeddable in nilpotent groups is followed in obtaining a finite set of implications which defines $({\cal CS}({\cal N}\sb{n}))\sb{s}.$ In addition, a structural description of the members of $({\cal CS}({\cal N}\sb{n}))\sb{s}$ is obtained. In a broader context, given a variety ${\cal V}$ of bands which contains the variety of all semilattices, a finite set of implications is presented to define the quasivariety of all semigroups that are embedable in ${\cal V}$-bands of abelian groups.
This paper has been withdrawn.