Varieties of structurally trivial semigroups. I
For any semigroup \(S\) and for any pair \((n,m)\) of non-negative integers, the author defines the relation \(\theta(n,m)\) on \(S\) as the set of all pairs \((a,b)\in S\times S\) such that \(aub=avb\) 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 \(V^{(n,m)}=\{S:S/\theta(n,m)\in V\}\) is again a variety and the mapping \(V\mapsto V^{(n,m)}\) is shown to be an injective endomorphism of the lattice of all semigroup varieties. 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 (\(k=1,2,\ldots\)). He then establishes several relations (inclusions and equalities) between varieties in \(\wp\), their lattice joins and meets. If \(V\) is a variety of structurally trivial semigroups then the set of varieties from \(\wp\) contained in \(V\) is called the skeleton of the lattice \(L(V)\) of all subvarieties of \(V\). The main result of the paper (Theorem 6.1) consists in a complete description of the skeleton of the lattice \(L(RB^3)\) where \(RB^3\) is the variety of all 3-nilpotent extensions of a rectangular band. However, an example shows that there are certain subvarieties of \(RB^3\) beyond this skeleton.
nilpotent extensions of rectangular bands
scientific article; zbMATH DE number 1251280