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

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... 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 [12] ). 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 [14] .