Certain operators related to Mal'cev products on varieties of completely regular semigroups
Semigroups which are unions of groups are said to be completely regular (c.r.). It is well-known that the class \({\mathfrak C}{\mathfrak R}\) of all c.r. semigroups is a variety of unary semigroups. Given subvarieties \(\mathfrak A\) and \(\mathfrak B\) of \({\mathfrak C}{\mathfrak R}\), their Mal'cev product is denoted \({\mathfrak A} \cdot {\mathfrak B}\). In general \({\mathfrak A} \cdot {\mathfrak B}\) is a quasi-variety and need not be a variety. We denote the subvariety of \({\mathfrak C}{\mathfrak R}\) generated of \({\mathfrak A} \cdot {\mathfrak B}\) by \({\mathfrak A} \odot {\mathfrak B}\). For a variety \(\mathfrak X\) of c.r. semigroups, let \({\mathcal L}({\mathfrak X})\) be the lattice of its subvarieties. If \(\Sigma\) is a system of identities then \([\Sigma]\) denotes the variety of c.r. semigroups defined by \(\Sigma\). The following notations are used in this paper: \({\mathfrak B} = [x = x^ 2]\), \({\mathfrak L}{\mathfrak Z} = [xy = x]\), \({\mathfrak R}{\mathfrak Z} = [xy = y]\), \({\mathfrak R}{\mathfrak B} = [xyx = x]\). Four operators on \({\mathcal L}({\mathfrak C}{\mathfrak R})\) are defined as follows: for any \({\mathfrak A} \in {\mathcal L}({\mathfrak C}{\mathfrak R})\) \({\mathfrak A}L_ l [L_ r,L,L^*] = \{S \in {\mathfrak C}{\mathfrak R} : eS [Se,eSe,eSf] \in {\mathfrak A}\) for all \(e,f \in E(S)\}\). The operators \(L_ l\), \(L_ r\), \(L\), \(L^*\) are used to characterize the variety \({\mathfrak A} \odot {\mathfrak B}\) for some \({\mathfrak B}\). Namely for any \({\mathfrak A} \in {\mathcal L}({\mathfrak C}{\mathfrak R})\): \({\mathfrak A}L_ l = {\mathfrak A} \odot {\mathfrak L}{\mathfrak Z}\), \({\mathfrak A}L_ r = {\mathfrak A} \odot {\mathfrak R}{\mathfrak Z}\), \({\mathfrak A}L^* = {\mathfrak A} \odot {\mathfrak B}\). Further, it is proved that the equivalence relations on \(L({\mathfrak C}{\mathfrak R})\) induced by \(L_ l\) or \(L_ r\) are complete congruences on \({\mathcal L}({\mathfrak C}{\mathfrak R})\) and the behaviour of \(L_ l\) or \(L_ r\) on \({\mathcal L}({\mathfrak B})\) is completely determined. At last, it is shown that \(L = L_ l L_ r = L_ r L_ l = L^*\). The paper contains some results on properties of the operations \(\cdot\) and \(\odot\) on \({\mathcal L}({\mathfrak C}{\mathfrak R})\). Notice that some results on \({\mathcal L}({\mathfrak B})\) follow easily from the description of the groupoid \(({\mathcal L}({\mathfrak B}),\odot)\) obtained by \textit{E. V. Sukhanov} [Semigroup Forum 14, No. 2, 143-150 (1977; Zbl 0366.20043)]. Unfortunately, the author does not refer to this work.
scientific article; zbMATH DE number 645078