\(E^{\vee}\)-unitary covers for \(\vee\)-semilatticed inverse semigroups (Q1941738)

Let \(S\) be a semigroup and \(\leq\) be a partial order on \(S\) which is compatible with the multiplication. \(S\) is called \(\vee\)-semilatticed provided that \(a\vee b\) exists for any \(a,b\in S\) and the multiplication distributes over \(\vee\). Natural example of \(\vee\)-semilatticed inverse semigroups are the semigroups \(K(G)\) (where \(G\) is a group) of all right cosets of various subgroups of \(G\), where the order is the subset inclusion. The semigroups \(K(G)\) play a role in the work of \textit{D. B. McAlister} and \textit{N. R. Reilly} [Pac. J. Math. 68, 161--174 (1977; Zbl 0368.20043)]. The current paper investigates the question of existence and description of \(E^{\vee}\)-unitary covers of \(\vee\)-semilatticed inverse semigroups. The results imply that not every \(\vee\)-semilatticed inverse semigroup admits a \(E^{\vee}\)-unitary cover. Also, free \(\vee\)-semilatticed inverse semigroups on more than one generator are not \(E\)-unitary. Particular important classes of \(\vee\)-semilatticed semigroups are naturally, co-naturally and lattice-ordered semigroups. Sufficient conditions of existence of \(E^{\vee}\)-unitary covers are obtained for these classes.