Structure of radicals in bands of semigroups (Q1179485)

Let \(R\) be a semigroup radical, \(B\) a band, and \(S=\bigcup_{b\in B}S_ b\) a band of semigroups. Denote by \(R(S,B)\) the congruence of \(S\) generated by all the congruences contained in \(\bigcup_{b\in B}R(S_ b)\). We say that \(R\) is restorable by the components of \(B\) if \(R(S)=R(S,B)\) for every band of semigroups \(S=\bigcup_{b\in B}S_ b\). The author gives necessary and sufficient conditions on a band \(B\) for the Jacobson, the Baer, the Brown-McCoy, and the least special radical to be restorable by the components of \(B\). These results were announced without proof [in Sov. Math. 32, No. 12, 119-123 (1988); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1988, No. 12(319), 71-73 (1988; Zbl 0684.20051)]. Similar results for the class of semigroups with zero can be found [in Semigroup Forum 38, 57-76 (1989; Zbl 0661.20043)].