On F-inverse covers of inverse monoids

Abstract

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis-Meakin expansion M(H ) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin-Zelmanov varieties K n of all locally finite groups of exponent dividing n.

F-inverse covers of E-unitary inverse monoids

“…This shows that the study of the F -inverse covers of nite-above inverse monoids can be reduced to the study of the F -inverse covers of nite-above E-unitary inverse monoids in the same way as in the case of nite inverse monoids generated by their maximal elements, see [2]. Furthermore, the fundamental observations [2, Lemmas 2.3 and 2.4] can be easily adapted to quasi-A-generated nite-above inverse monoids.…”

“…We are ready to describe the graph condition Auinger and Szendrei have introduced in their paper [2] as a reformulation of the F -inverse cover problem. Their key step is the assertion that every nite inverse monoid admits a nite F -inverse cover if and only if, for every nite connected graph Γ, there exist a locally nite group variety U and a dual premorphism ψ :…”

“…In [2], C 0 (x) is incorrectly dened to be the graph spanned by the U-content of x together with ιx. From the proof of [2, Lemma 3.1] (see the inclusion µ(xψ) ⊆ C 0 (x)), it is clear that the denition of C 0 (x) needed is the one in (2.3.2).…”

“…In the following chapter, we develop a framework analogous to the one in [2], which allows us to investigate the proposed problem for a large class of E-unitary inverse monoids. First, we dene the class of E-unitary inverse monoids we intend to consider.…”

“…The most important antecedent to the research presented in the dissertation is the paper of Auinger and Szendrei [2] on the question. They go a step further by applying that it is sucient to restrict to a special class of E-unitary inverse monoids called MargolisMeakin expansions, which, as we will see, have a very convenient structure.…”