Cryptic Semigroup Varieties

Abstract

We describe all minimal noncryptic periodic semigroup [monoid] varieties. We prove that there are exactly three distinct maximal cryptic semigroup [monoid] varieties contained in the variety determined by xn ≈ x n+m, n ≥ 2, m ≥ 2. Analogous results are obtained for pseudovarieties: there are exactly three maximal cryptic pseudovarieties of semigroups [monoids]. It is shown that if a cryptic variety or pseudovariety of monoids contains a nonabelian group, then it consists of bands of groups only. Several characterizations are given of the cryptic overcommutative semigroup [monoid] varieties.