Bivarieties of orthodox semigroups generated by generalized Mal'cev products
The notion of Malcev product, originally defined in the context of varieties, has been extended to e-varieties of regular semigroups (classes closed under direct products, quotients and regular subsemigroups; sometimes called ``bivarieties'', as by the author): if \({\mathcal U},{\mathcal V}\) are e-varieties, then \(\langle{\mathcal U}{\mathbf m}{\mathcal V}\rangle\) is the e-variety generated by the regular semigroups \(S\) possessing a congruence whose quotient belongs to \(\mathcal V\) and such that the congruence classes of individual idempotents are semigroups belonging to \(\mathcal U\). The author introduces a variant, termed the generalized Malcev product: \(\langle{\mathcal U}{\mathbf g}{\mathbf m}{\mathcal V}\rangle\) is defined similarly to \(\langle{\mathcal U}{\mathbf m}{\mathcal V}\rangle\), except that only the regular elements of the congruence classes of idempotents (which in this context always form semigroups) need belong to \(\mathcal U\). An example is given to show that the products are different. It is also interpreted in terms of regular relational morphisms. Some interesting e-varieties are shown to arise in this way. For instance, if \(\mathcal U\) is any orthodox e-variety and \({\mathcal L}{\mathcal Z}\) denotes the (e-)variety of left zero semigroups, then \(\langle{\mathcal U}{\mathbf g}{\mathbf m}{\mathcal L}{\mathcal Z}\rangle\) comprises the orthodox semigroups with the property that the regular elements of each principal right ideal belong to \(\mathcal U\).
scientific article; zbMATH DE number 1057432