On embeddability into a semidirect product of a band by a group
Let \(S\) be a given orthodox semigroup. The band of idempotents is denoted by \(E_S\), the least group congruence of \(S\) is denoted by \(\sigma_S\). A regular semigroup \(S\) is called \(E\)-unitary if \(E_S\) forms a \(\sigma_S\)-class. \textit{M. B. Szendrei} [in Proc. R. Soc. Edinb., Sect. A 106, 89-102 (1987; Zbl 0628.20052)] raised the problem whether each \(E\)-unitary regular semigroup is embeddable into a semidirect product of a band by a group. Up to now many examples of embeddable \(E\)-unitary regular semigroups are known [see \textit{B. Billhardt, M. B. Szendrei}, Proc. Edinb. Math. Soc., II. Ser. 38, No. 3, 361-385 (1995; Zbl 0837.20070) and \textit{M. B. Szendrei}, Int. J. Algebra Comput. 3, No. 3, 317-333 (1993; Zbl 0796.20053)]. In the present paper the author solves the embeddability problem in the negative. An example of an \(E\)-unitary regular semigroup is presented which is not embeddable into a semidirect product of a band by a group. For each embeddable \(E\)-unitary regular semigroup \(S\) a canonical embedding into a homomorphic image of the free product of \(S\) and \(S/\sigma_S\) is constructed. This canonical embedding is of a purely algebraic nature.
scientific article; zbMATH DE number 1214141