Restriction \(\omega \)-semigroups
One of the classic theorems of inverse semigroup theory, found independently by \textit{W. D. Munn} [Glasg. Math. J. 9, 46--66 (1968; Zbl 0165.33504)] and \textit{B. P. Kochin} [Vestn. Leningr. Univ., Mat. Mekh. Astron. 23, No. 2, 41--50 (1968; Zbl 0162.32903)], determines the simple inverse semigroups (in fact, by means of a modest extension, all such inverse semigroups), whose semilattice of idempotents is an $\omega$-chain, that is, isomorphic to the nonnegative integers under the reverse of the usual order. The characterization is by ``Bruck-Reilly'' semigroups over finite chains of groups. Various authors have extended this theory (and its precursor, for bisimple $\omega$-semigroups) to classes of ample semigroups and, then, of restriction semigroups. In such semigroups, the semilattice of all idempotents is replaced by a semilattice $E$ of ``distinguished'' idempotents, or ``projections''. The corresponding generalized Green's relations $\widetilde{\mathcal{R}}$, $\widetilde{\mathcal{L}}$, and so on, may then be used to define $\sim$-bisimplicity and $\widetilde{\mathcal{J}}$-simplicity, by analogy. In the class of restriction semigroups, the natural generalization of a finite chain of groups is the strong semilattice of a finite chain of monoids. The authors show that Bruck-Reilly semigroups over such semilattices of monoids are $\widetilde{\mathcal{J}}$-simple restriction $\omega$-semigroups. However, they have the additional property of possessing an ``inverse skeleton'': each member $a$, say, of an $\widetilde{\mathcal{H}}$-class is ``$E$-regular'', that is, possesses an inverse $a^0$ such that $aa^0$, $a^0a$ belong to $E$. The main result is then that the $\widetilde{\mathcal{J}}$-simple restriction $\omega$-semigroups having an inverse skeleton correspond precisely to the above Bruck-Reilly semigroups, generalizing the Munn-Kocin theorem. Reducing to the $\sim$-bisimple case yields the theorem of \textit{S. Ma} et al. [Front. Math. China 8, No. 6, 1391--1405 (2013; Zbl 1309.20047)]. Notice that the upper and lower ``halves'' of the bicyclic semigroup are $\widetilde{\mathcal{J}}$-simple restriction $\omega$-semigroups without an inverse skeleton.
$\widetilde{\mathcal{J}}$-simple restriction $\omega$-semigroups
scientific article; zbMATH DE number 7031581
scientific article