Rectangular group congruences on a semigroup. (Q372337)

A semigroup \(S\) is said to be a rectangular group if it is isomorphic to the direct product \(G\times M\) of a group \(G\) and a rectangular band \(M\). In this paper, the author studies rectangular group congruences on an arbitrary semigroup. Some results of this paper are an extension of the results obtained by \textit{F. E. Masat} [Proc. Am. Math. Soc. 50, 107-114 (1975; Zbl 0317.20038)]. A rectangular group congruence is called \textit{proper} if it is neither a group nor a matrix congruence on \(S\). The author first gives necessary and sufficient conditions on a semigroup \(S\) which has a proper rectangular group congruence. Furthermore, the author shows that every rectangular group congruence on \(S\) is the intersection of a group congruence and a matrix congruence. In addition, if \(S\) is \(E\)-inversive, then this expression is unique. Moreover, each rectangular group congruence on an \(E\)-inversive semigroup is uniquely determined by its kernel and trace.