The lattice of varieties of completely regular semigroups.
The completely regular semigroups form a variety \(\mathcal{CR}\), when considered as unary semigroups. After a period in which various principal ideals of the lattice \({\mathcal L}(\mathcal{CR})\) were described, \textit{L. Polák} [Semigroup Forum 36, No. 3, 253-284 (1987); ibid. 37, No. 1, 1-30 (1988; Zbl 0638.20032)] made a breakthrough with a theorem that provided a framework for studying the lattice as a whole. (To be precise, the theorem considered only the interval above the variety \(\mathcal{SL}\) of semilattices. Since the complement of this interval consists of the varieties of completely simple semigroups, in the opinion of many this restriction is of no consequence.) Each variety is represented by a `ladder' labelled by elements of the lattice \(\mathcal K\), where \(\mathcal K\) comprises the set \({\mathcal K}_0\) of minimum elements \({\mathcal V}_K\) of the \(K\)-classes of \({\mathcal L}(\mathcal{CR})\) -- where \(K\) is a certain well known congruence on the lattice -- with three additional elements adjoined at its bottom. While in its most general form this theorem can hardly be expected to completely describe the entire lattice, its usefulness is illustrated in the case of orthogroups, for instance, where the lattice \(\mathcal K\) is essentially the lattice of group varieties, and the theorem may be regarded as providing an immediate, effective description of the lattice of orthogroup varieties. In the present paper, the author provides an alternative view, this time of the entire lattice \({\mathcal L}(\mathcal{CR})\), taking advantage of the well studied lattice of band varieties as a way of replacing the `ladder' aspect of Polák's description. With each completely regular variety \(\mathcal V\) is now associated the obvious band variety, the variety \({\mathcal V}_K\) and a sequence of members of \({\mathcal K}_0\). Once again, it is shown how this theorem specializes and simplifies in the case of some well known varieties of completely regular semigroups.
scientific article; zbMATH DE number 5009760