The G-Exponent of a Pseudovariety of Semigroups,

Ashʹs proof of the Pointlike Conjecture provides an algorithm for calculating the group-pointlike subsets of a finite semigroup S. We denote by PG(S) the subsemigroup of P(S), the elements of which are the group-pointlike subsets of S. This paper is concerned with some properties of the operator G, which assigns to the pseudovariety V the pseudovariety GV generated by the semigroups PG(S) with S in V. As GV is a subpseudovariety of V, some of them come from properties of the power operator. We show that 3GV = S for any pseudovariety of semigroups V that contains a semigroup such that the subsemigroup generated by its idempotents is non-permutative.