On \(F\)-inverse covers of finite-above inverse monoids. (Q5964490)

An inverse monoid \(M\) is called: 1) \textit{finite above} if the set \(m^\omega=\{n\in M:n\geq m\}\) is finite for every \(m\in M\) and 2) \textit{F-inverse} if each class of the least group congruence has a greatest element with respect to the natural partial order. Let \(v\in M/\sigma\) be a \(\sigma\)-class (\(\sigma\) is a smallest group congruence on \(M\)) and let \(h(a,b;v)\) be a least upper bound of the set of idempotents \(\{d^{-1}ab^{-1}d:d\in\max v\}\) with \(a\sigma b\) (\(\max v\) denotes the set of maximal elements of \(v\)). Now (C) denotes the following condition: \(c\cdot h(a,b;v)\cdot c^{-1}b\not\leq a\) for some \(c\in\max v\). If \(M\) is generated by \(A\cup E(M)\) for a subset \(A\subseteq M\) then \(M\) is called \textit{quasi-\(A\)-generated}.NEWLINENEWLINE Four equivalent conditions for a quasi-\(A\)-generated finite-above \(E\)-unitary inverse monoid to have an \(F\)-inverse cover via a group variety are found. Restricting to the varieties of Abelian groups, it is proved that if \(M\) is a finite-above \(E\)-unitary inverse monoid such that for some \(a,b\in\max M\) with \(a\sigma b\) and for some \(v\in M/\sigma\), condition (C) is satisfied, then \(M\) has no \(F\)-inverse cover via Abelian groups. An example is constructed showing that for any group \(G\), there exist finite-above \(E\)-unitary inverse monoids with greatest group homomorphic image \(G\) that fail to be \(F\)-inverse but admit \(F\)-inverse covers via Abelian groups.