Varieties of structurally trivial semigroups. II.
For any semigroup \(S\) and for any pair \((n,m)\) of non-negative integers, let \(\theta(n,m)=\{(a,b)\in S\times S\mid aub=avb\) for all \(u\in S^n\), \(v\in S^m\}\). This relation is always a congruence on \(S\). A semigroup \(S\) is called structurally trivial if \(\theta(n,m)=S\times S\) for some \(n\) and \(m\). This is equivalent to \(S\) being a nilpotent extension of a rectangular band. If \(V\) is a variety of semigroups then the class \(V^{(n,m)}=\{S\mid S/\theta(n,m)\in V\}\) is again a variety. The author denotes by \(\wp\) the set of all varieties of the form \(W^{(n,m)}\) where \(W\) is either the variety of all right zero bands, or the variety of all left zero bands, or the variety of all \(k\)-nilpotent semigroups, or the variety of all \(k\)-nilpotent extensions of a rectangular band (\(k=1,2,\dots\)). He proves that \(\wp\) is a distributive sublattice of the lattice of all semigroup varieties and provides a complete description of this sublattice. The sublattice \(\wp\) contains sublattices described earlier by \textit{I. I. Mel'nik} [Izv. Vyssh. Uchebn. Zaved., Mat. 1971, No. 12(115), 103-108 (1971; Zbl 0241.20047)], \textit{Yu. G. Koshelev} [Semigroup Forum 12, 95-107 (1976; Zbl 0338.20095)], and the author [part I, Semigroup Forum 58, No. 2, 159-174 (1999; Zbl 0922.20063)].
nilpotent extensions of rectangular bands
scientific article; zbMATH DE number 1972502