On special types of Putcha semigroups whose subgroups belong to a given variety of groups. (Q2901413)

A semigroup such that \(a\) divides \(b\) implies \(a^2\) divides some power of \(b\) is said to be a `Putcha semigroup' in honor of \textit{M. S. Putcha} [Semigroup Forum 6, 12-34 (1973; Zbl 0256.20074)], who showed that this property characterizes the semilattices of Archimedean semigroups. Given a variety \(\mathbf H\) of groups, a congruence on a semigroup is said to be an `\(\mathbf{LH}\)-congruence' if the local submonoids of every idempotent class are groups from \(\mathbf H\).NEWLINENEWLINE The main result in the paper states that, for a semilattice of ideal extensions of completely simple semigroups, the subgroups belong to \(\mathbf H\) if and only if the least semilattice congruence is a maximal \(\mathbf{LH}\)-congruence. This result applies in particular to periodic Putcha semigroups with only finitely many idempotents and to periodic permutative semigroups. In the finite case, the author also identifies the least semilattice congruence of a Putcha semigroup with its Rhodes radical, using a result of the reviewer et al. [Trans. Am. Math. Soc. 361, No. 3, 1429-1461 (2009; Zbl 1185.20058)].