Operators and products in the lattice of existence varieties of regular semigroups
The authors introduced several complete congruences on the lattice \(L={\mathcal L}_{ev}({\mathcal R}{\mathcal S})\) of e- (or ``existence'') varieties of regular semigroups [in J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051)], generalizing the corresponding notions for the lattice of varieties of completely regular semigroups. Whilst in the latter case these congruences have natural syntactic descriptions, this cannot be so in general, for free regular semigroups do not exist. Thus alternative interpretations are desirable. Since the cited paper was written, various authors, including K. Auinger and T. E. Hall, have in fact generated further such congruences, together with some such interpretations. The purpose of the current paper is to provide descriptions of four of the original congruences in terms of semidirect products. Given any complete congruence \(\rho\) on the lattice \(L\), and an e-variety \(\mathcal U\), the \(\rho\)-class \({\mathcal U}\rho\) is an interval, of the form \([{\mathcal U}_\rho,{\mathcal U}^\rho]\). For \(\rho=T_l\) or \(K_l\), \({\mathcal U}^\rho\) is [loc. cit.] the Malcev product \({\mathcal L}{\mathcal G}\circ{\mathcal U}\) or \({\mathcal L}{\mathcal Z}\circ{\mathcal U}\), respectively, where \({\mathcal L}{\mathcal G}\) and \({\mathcal L}{\mathcal Z}\) are the e-varieties of left groups and of left zero semigroups, respectively. There are natural duals. (Alternatively, these formulas may be used to define the respective congruences). The relationship between Malcev products and semidirect products has a long history for varieties of groups and for varieties, and pseudovarieties, of semigroups. Thus it is natural to look for descriptions in terms of semidirect (or wreath) products. One of the main results of the paper is that for any nongroup e-variety \(\mathcal U\), \({\mathcal U}^{T_l}={\mathcal G}*{\mathcal U}\), where \({\mathcal G}\) is the (e-) variety of groups and \({\mathcal G}*{\mathcal U}\) is the e-variety generated by all semidirect products \(G*U\), for \(G\in{\mathcal G}\), \(U\in{\mathcal U}\). More generally, if \(\mathcal H\) is any nontrivial group variety, then \({\mathcal H}*{\mathcal U}\) is the e-variety generated by the Malcev product \(({\mathcal L}{\mathcal Z}\vee{\mathcal H})\circ{\mathcal U}\). (\textit{P. G. Trotter} and the reviewer [Semidirect products of regular semigroups (Trans. Am. Math. Soc., to appear)] have introduced a more general semidirect product \({\mathcal U}*{\mathcal V}\), defined whenever either \({\mathcal U}\) or \(\mathcal V\) consists of completely simple semigroups. They proved a weaker version of the above result.) The e-variety \({\mathcal U}^{K_l}\) is shown to be representable in a similar way, by replacing the usual semidirect product by a certain subsemigroup, the first factors this time being semilattices. The dual congruences \(T_r\) and \(K_r\) are treated using the reverse semidirect products.
lattice of varieties of completely regular semigroups
scientific article; zbMATH DE number 895847
scientific article