In this paper I prove that if a semigroup S is stable then [formula omitted] L(S) and [formula omitted] R(S) (the Rhodes expansions), and [formula omitted]+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function. The techniques used were originally developed by the author to prove that [formula omitted]+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).
All Science Journal Classification (ASJC) codes