\(TK\)-operator semigroups for cryptogroups (Q1976427)

A by now standard way of studying the lattice of congruences on a regular semigroup is via the trace and kernel operators: with any congruence \(\rho\) are associated \(\rho k\), \(\rho K\), \(\rho t\), \(\rho T\), respectively the least and greatest congruences with the same kernel (union of idempotent classes) as \(\rho\), and the least and greatest congruences with the same trace (restriction to the idempotents) as \(\rho\). Regarded as operators on the lattice of congruences, \(k,K,t,T\) generate a semigroup, its ``\(TK\)-operator semigroup''. First studied by \textit{M.~Petrich} and \textit{N.~R.~Reilly} [Trans. Am. Math. Soc. 270, 309-325 (1982; Zbl 0484.20026)], in the context of inverse semigroups, this operator semigroup has been explicitly computed in several concrete situations. The purpose of the present paper is to extend work of \textit{M.~Petrich} [J. Aust. Math. Soc., Ser. A 56, No. 2, 243-266 (1994; Zbl 0807.20050)] on this operator semigroup from completely simple and Clifford semigroups to cryptogroups, also commonly known as bands of groups. The most general such semigroup is explicitly constructed and turns out to be the operator semigroup exhibited by Petrich for completely simple semigroups. Various special cases are considered, such as for \(E\)-unitary cryptogroups and certain subclasses of completely simple semigroups.