Minimal noncryptic e-varieties of regular semigroups
A regular semigroup \(S\) is cryptic if Green's relation \(\mathcal H\) is a congruence on \(S\); an e-variety \(\mathcal V\) is cryptic if each member of \(\mathcal V\) happens to be so. Cryptic e-varieties of completely regular respectively inverse semigroups have been completely classified by \textit{V. V. Rasin} [Semigroup Forum 23, 201-220 (1981; Zbl 0479.20028)] respectively \textit{N. R. Reilly} [Q. J. Math., Oxf. II. Ser. 36, 467-487 (1985; Zbl 0582.20038)] in the language of ``forbidden members'' (thereby giving, at the same time, a complete list of minimal non-cryptic e-varieties of completely regular respectively inverse semigroups). The paper under review presents a highly non-trivial extension of these results to all e-varieties of regular semigroups: an e-variety is cryptic if and only if it does not contain a member of either of the lists of Rasin and Reilly. In other words, an e-variety is cryptic if and only if its intersections with completely regular semigroups and inverse semigroups happen to be cryptic. As a further result, all e-varieties all of whose members are group closed are characterized simply by adding to the mentioned list of forbidden members the 5-element non-orthodox completely 0-simple semigroup \(C_2\). Recall that a semigroup is group closed if its group elements form a subsemigroup.
scientific article; zbMATH DE number 941271