A finiteness condition for semigroups generalizing a theorem of Coudrain and Schützenberger (Q1341145)

Let \(T'\) be a subsemigroup of a semigroup \(T\). If a) \(T\) satisfies the minimum condition for bi-ideals \((\min_B)\) and b) every subgroup of \(T\) generated by some finite subset of \(T'\) is finite then \(T'\) is locally finite. The case \(T' =T\) yields a generalization of a theorem of \textit{M. Coudrain} and \textit{M. P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. A 262, 1149-1151 (1966; Zbl 0141.01801)]. Also, a new proof of the theorem of \textit{R. McNaughton} and \textit{Y. Zalcstein} [J. Algebra 34, 292-299 (1975; Zbl 0302.20054)] based on this result is given.