### Stability and J-depth of expansions

Jean-Camille Birget
1988 Bulletin of the Australian Mathematical Society
In this paper I prove that if a semigroup S is stable then A^(S) and AJJ(S) (the Rhodes expansions) , and A-|-(S^4) (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 «/*-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
more » ... a J-depth function. The techniques used were originally developed by the author to prove that A-|-(Sjt) is finite if S is finite (later Rhodes found a much more direct proof of that result). Importance of the notions of stability and J-depth. Stability is a condition in Rees' theorem. The J-depth is needed for carrying out decompositions of a semigroup (for example to prove global theorems in the style of Krohn-Rhodes). One can also view stability and existence of the J-depth function as a generalisation of finiteness: many theorems about finite semigroups carry over nicely in this case. Stability is a generalisation of torsion (every torsion semigroup is stable). Torsion by itself is not a good enough generalisation of finite, being too much a local property. Stability is a "locaf property, in the sense that if refers only to each /-class separately. On the other hand, existence of the J-depth function is a purely global property of the J-order (which ignores the inside of the /-classes). Semigroups that are stable and have a J-depth function arise for example as limits of finite semigroups (see  ). This approach might be useful in the study of models of computation, especially parallel computation. Structure theorem for semigroups that are stable and have a J-depth function. Such semigroups have a structure theorem (generalising the case of finite semigroups) which combines Rees' theorem and the Krohn-Rhodes theorem. In the finite case that theorem was first stated and proved by Rhodes and Allen  .