Semigroups satisfying some variable identities (Q2709035)

Let \(A^+\) be the free semigroup over an alphabet \(A\), \(\Sigma\) be a set of identities over \(A\). The authors say that a semigroup \(S\) satisfies \(\Sigma\) as variable identities if the kernel of each homomorphism from \(A^+\) into \(S\) contains a non-trivial identity from \(\Sigma\). This is the same concept as introduced by \textit{M. S. Putcha} and \textit{J. Weissglass} [in Semigroup Forum 3, 64-67 (1971; Zbl 0235.20055) and Trans. Am. Math. Soc. 168, 113-119 (1972; Zbl 0245.20051)], but the definition given here is closer to the definition of ordinary identities than the one of Putcha and Weissglass. In the above mentioned papers, Putcha and Weissglass used variable identities to characterise periodic semigroups which are nilpotent extensions of unions of groups and semilattices of groups. In the paper under review they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in general and in various special cases. The results obtained generalize those of Putcha and Weissglass, as well as results of \textit{H. E. Bell} [Pac. J. Math. 70, 29-36 (1977; Zbl 0364.16012)] concerning rings satisfying variable semigroup identities and the results of \textit{M. Ćirić} and \textit{S. Bogdanović} [Semigroup Forum 48, No. 3, 303-311 (1994; Zbl 0803.20037)] concerning semigroups satisfying some ordinary identities.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].