Malcev products of monoids and varieties of bands (Q520002)

The study is motivated by the fact that the class of completely regular semigroups can be written as a Malcev product \(({\mathcal G} \circ {\mathcal RB}) \circ {\mathcal S}\), where \({\mathcal G}\), \({\mathcal RB}\) and \({\mathcal S}\) are the varieties of groups, rectangular bands and semilattices, respectively. The author describes the structure of semigroups that belong to the different classes obtained when replacing \({\mathcal G}\) above by the class \({\mathcal M}\) of monoids. For classes of semigroups \({\mathcal X}\) and \({\mathcal Y}\), \({\mathcal X} \square{\mathcal S}\) denotes a special case of Malcev product -- the class of all strong semilattices of semigroups \([Y; S_{\alpha}, \chi_{\alpha,\beta}]\) where \(Y \in {\mathcal S}\) and \(S_{\alpha}\in {\mathcal X}\) for all \(\alpha \in Y\). A subclass of the latter consisting of all unary semigroups isomorphic to some \([Y; S_{\alpha}, \chi_{\alpha,\beta}]\) where \(\chi_{\alpha,\beta}\) is injective for all \(\alpha \geq\beta\) is denoted by \({\mathcal X} \bigtriangleup{\mathcal S}\). For nonempty sets \(I,\Lambda\) and a \(\Lambda \times I\)-matrix \(P\), the class of all semigroups isomorphic to Rees matrix semigroups \({\mathcal M}(I,T, \Lambda;P)\), \(T\in {\mathcal X}\), is denoted by \({\mathcal X} \ast {\mathcal RB}\). The author describes the elements of several classes including the following: \(({\mathcal G} \circ {\mathcal S}) \circ {\mathcal RB} (= {\mathcal G} \circ ({\mathcal S} \circ {\mathcal RB}))\), \({\mathcal M} \circ {\mathcal B}\) (\({\mathcal B}\) denotes the variety of bands), \({\mathcal M} \ast {\mathcal RB}\), \({\mathcal M} \square{\mathcal S}\), \(({\mathcal M} \circ {\mathcal RB}) \circ{\mathcal S}\), \(({\mathcal M} \circ {\mathcal RB}) \square{\mathcal S}\), \(({\mathcal M} \times {\mathcal RB}) \square{\mathcal S}\), \(({\mathcal M} \circ {\mathcal S}) \circ {\mathcal RB}\), \({\mathcal M} \circ ({\mathcal S} \circ {\mathcal RB})\), \(({\mathcal M} \times {\mathcal RB}) \bigtriangleup {\mathcal S}\). Based on the obtained results, a \(\cap\)-semilattice of the classes considered above is constructed. For example, it turns out that \(({\mathcal M} \circ {\mathcal B})\cap(({\mathcal M} \circ {\mathcal S}) \circ {\mathcal RB})={\mathcal M} \circ ({\mathcal S} \circ {\mathcal RB})\).