A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

Abstract

We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to compute whether a finite X-generated group H admits a canonical dual prehomomorphism into the Margolis–Meakin expansion M(G) of a finite X-generated group G. We show that for G the Klein four-group a suitable H must be of exponent 6 at least and recapture a result of Szakács.

Solution to the Henckell--Rhodes problem: finite $F$-inverse covers do exist

“…The importance of the latter paper went beyond its immediate task as in the following years interesting and deep connections with the profinite topology of the free group [21] and finite model theory [12] have been revealed and studied [1,2]. Yet the second stated problem which was called by the authors a "stronger form of the pointlike conjecture for inverse monoids" has not been solved in Ash's paper and has since then attracted considerable attention [14,25,26,5,24,23,8]. It asked: Problem 1.1.…”