THE RHODES EXPANSION AND FREE OBJECTS IN VARIETIES OF COMPLETELY REGULAR-SEMIGROUPS

[摘要] For any semigroup S, the Rhodes expansion Rh(S) of S consists of all finite sequences (a1,...,a(m)) of elements of S such that a1 < L a2 < L...< L a(m) and with the multiplication given by (a1,...,a(m))(b1,...,b(n)) = red(a1b1,...,a(m)b1,b1,b2,...,b(n)) where, for any such sequence (c1,...,c(k)), red(c1,...,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-epsilon-A. If V is any variety of completely regular semigroups that contains the variety of semilattices and if FV is the free object in V, then Rh(x)(FV(x)) is the free object on X in the variety RZ o V, where RZ denotes the variety of right zero semigroups and o denotes the Mal'cev product of varities (which, in this case yields a variety).

[发布日期] 1990-12-20