Bicyclic extensions (Q1271932)

The author considers a more general structure theory of simple semigroups, the basic idea being to give a construction where the term ``group'' is replaced by the term ``monoid'' [see: \textit{D. Rees}, Q. J. Math., Oxf. Ser. 19, 101-108 (1948; Zbl 0030.00802)]. The resulting construction is called a bicyclic extension. In this paper, a characterization of bicyclic extensions of finite chains of groups, which generalizes Kochin's characterization of simple \(\omega\)-inverse semigroups [\textit{B. P. Kochin}, Vestn. Leningr. Univ. 23, No. 7, Ser. Mat. Mekh. Astron. No. 2, 41-50 (1968; Zbl 0162.32903)] is given. These extensions characterize simple regular semigroups \(S\) such that the set of idempotents of \(S\) is a strict \(\omega\)-chain of bands \(P_n\times Y\), \(n\in\mathbb{N}\), where \(P_n\) is a right zero semigroup (\(xy=y\)) and \(Y\) is a finite chain.