Characterizing some completely regular semigroups by their subsemigroups. (Q2861589)

The author considers several varieties of completely regular semigroups such as groups, completely simple semigroups, semilattices, semilattices of groups, cryptogroups (bands of groups), normal cryptogroups, orthogroups, and local orthogroups [see \textit{M. Petrich} and \textit{N. R. Reilly}, Completely regular semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts. 23. Chichester: Wiley (1999; Zbl 0967.20034)]. As the author writes in the abstract ``For each of these varieties we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties \(\mathcal V\) we determine minimal non-\(\mathcal V\) varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups.'' The main tool for study of cryptogroups is the construction of all completely regular monoids with two generators accomplished by \textit{M. Petrich} [J. Aust. Math. Soc. 90, No. 2, 271-287 (2011; Zbl 1228.20050)]. The final section consists of a review of the results obtained in the form of three tables and a figure.