A concept of variety for regular biordered sets
A regular biordered set \((E,\cdot,\omega^ l, \omega^ r)\) is a set \(E\) together with two quasiorders \(\omega^ l\) and \(\omega^ r\) and a partial binary operation \(\cdot\), satisfying a finite (but somehow complicated) system of axioms. Regular biordered sets were introduced by \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)] and are an abstract characterization of the partial groupoid \(E(S)\) of idempotents of a regular semigroup \(S\). Nambooripad also introduced suitable concepts of morphic image and substructure of regular biordered sets (called ``regular bimorphic image'' and ``relatively regular biordered subset''). In the paper under review, a concept of variety for regular biordered sets is introduced by putting a class \(\mathcal E\) of regular biordered sets to be a ``bivariety'' if \(\mathcal E\) is closed under taking direct products (which are defined in an obvious way), relatively regular biordered subsets, and regular bimorphic images. In the main theorem it is shown that the mapping \({\mathcal V} \mapsto \{E(S) \mid S \in {\mathcal V}\}\) is a complete surjective morphism from the lattice of all existence varieties of regular semigroups onto the lattice of all bivarieties of regular biordered sets. Another interesting result of the paper states that a bivariety \(\mathcal E\) of regular biordered sets contains all free objects if and only if \(\mathcal E\) consists entirely of local semilattices or solid biordered sets. Precisely in these cases, regular biordered sets can be equipped with a total binary operation (extending the partial biordered set product) so that the bivarieties are precisely the varieties of the resulting algebras.
lattice of existence varieties of regular semigroups
lattice of bivarieties of regular biordered sets
scientific article; zbMATH DE number 682638
scientific article