The Rhodes expansion and free objects in varieties of completely regular semigroups

Abstract

For any semigroup S, the Rhodes expansion Rh( S) of S consists of all finite sequences ( a 1,…, a m ) of elements of S such that a 1 < L a 2 < L ⋯ < L a m and with the multiplication given by ( a 1, …, a m )( b 1, …, b n ) = red( a 1 b 1,…, a m b 1, b 1, b 2,…, b n ) where, for any such sequence ( c 1,…, c k ), red( c 1,…, c k ) is the sequence obtained by deleting all but the leftmost element of L -equivalent elements. For any set A of generators of S, the reduced Rhodes expansion Rh A ( S) is the subsemigroup of Rh( S) generated by elements of the form ( a), a ∈ A. If V is any variety of completely regular semigroups that contains the variety of semilattices and if F V is the free object in V , then Rh X(F V X) is the free object on X in the variety RZ ∘ V , where RZ denotes the variety of right zero semigroups and ∘ denotes the Mal'cev product of varieties (which, in this case yields a variety).