Clusters of varieties of completely regular semigroups
Let \(\mathcal{CR}\) be the variety of completely regular semigroups equipped with the unary operation of inversion and let \(\mathcal{L(CR)}\) be its lattice of subvarieties. The relations \(\mathbb B^\wedge\) and \(\mathbb B^\vee\) on \(\mathcal{L(CR)}\) are defined by \(\mathcal U\mathbb B^\wedge\mathcal V\) if \(\mathcal U\cap\mathcal B=\mathcal V\cap \mathcal B\) and \(\mathcal U\mathbb B^\vee\mathcal V\) if \(\mathcal U\cup\mathcal B=\mathcal V\cup \mathcal B\) (\(\mathcal B\) the variety of bands) are complete congruences such that all of whose classes are intervals. The lower ends \(\mathcal V_{B^\wedge}\) and \(\mathcal V_{B^\vee}\) and the upper ends \(\mathcal V^{B^{\wedge}}\) and \(\mathcal{V}^{B^{\vee}}\) of these intervals induce four operators on \(\mathcal{L(CR)}\). The \textit{cluster at} \(\mathcal{V}\) is the following subset of \(\mathcal{L(CR)}\): cl(\(\mathcal{V})=\{\mathcal{V}_X, \mathcal{V}^Y\mid X, Y\in \{ \mathbb{B}^{\wedge}, \mathbb{B}^{\vee} \}^+ \} \). A big number of properties of these operators and clusters are deduced. For example, \(\mathcal{V}_{B^{\vee}}\)\(^{B^{\wedge}}\hskip-0.1cm\) = \(\mathcal{V}_{B^{\vee}}\)\(^{B^{\wedge}}\hskip-0.1cm\)\(_{B^{\vee}}\). The monoid generated by these operators is determined and studied, including commutativity properties of the operators. Special attention is paid to the varieties containing all groups.
scientific article; zbMATH DE number 6865409