An upper bound for the power pseudovariety PCS

Abstract

It is a celebrated result in finite semigroup theory that the equality of pseudovarieties PG = BG holds, where PG is the pseudovariety of finite monoids generated by all power monoids of finite groups and BG is the pseudovariety of all block groups, that is, the pseudovariety of all finite monoids all of whose regular ${\fancyscript{D}}$-classes have the property that the corresponding principal factors are inverse semigroups. Moreover, it is well known that BG = JⓜG, where JⓜG is the pseudovariety of finite monoids generated by the Mal’cev product of the pseudovarieties J and G of all finite ${\fancyscript{J}}$-trivial monoids and of all finite groups, respectively. In this paper, a more general kind of finite semigroups is considered; namely, the so-called aggregates of block groups are introduced. It follows that the class AgBG of all aggregates of block groups forms a pseudovariety of finite semigroups. It is next proved that AgBG = JⓜCS, where JⓜCS is the pseudovariety of finite semigroups generated by the Mal’cev product of the pseudovarieties J and CS, whilst, this once, J stands for the pseudovariety of all finite ${\fancyscript{J}}$-trivial semigroups and CS stands for the pseudovariety of all finite completely simple semigroups. Furthermore, it is shown that the power pseudovariety PCS, which is the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups, has the property that ${{\bf PCS}\subseteq{\bf AgBG}}$. However, the question whether this inclusion is strict or not is left open.