Existence varieties with lattices of regular subsemigroups
For a regular semigroup, the set \(\text{Sub}(S)\) of its regular subsemigroups forms a partially ordered set under set inclusion. In general, \(\text{Sub}(S)\) is not a sublattice of the lattice of all subsemigroups of \(S\); in fact, it need not even be a lattice at all. It is proven here that if \(S\) is either locally inverse or \(S\)-solid, then \(\text{Sub}(S)\) is indeed a complete lattice, the meet of a family of regular subsemigroups being the set of regular elements of their intersection, in either case. In the context of \(e\)-varieties (classes of regular semigroups closed under products, quotients and regular subsemigroups), this is best possible: any \(e\)-variety \(\mathcal U\) with the property that \(\text{Sub}(S)\) is a lattice for each \(S \in {\mathcal U}\) must be contained in either the \(e\)-variety of locally inverse regular semigroups or that of \(E\)-solid regular semigroups. \{Reviewer's remark: A similar result is proven by \textit{Y. T. Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)], where it is used to prove the existence of ``\(e\)-free'' objects in the \(e\)-varieties of locally inverse and \(E\)-solid semigroups\}.
scientific article; zbMATH DE number 488262
scientific article