We verify the "Type II Conjecture" concerning the question of which elements of a finite monoid M are related to the identity in every relational morphism with a finite group. We confirm that these elements form the smallest submonoid, K, of M (containing 1 and) closed under "weak conjugation", that is, if x ∈ K, y ∈ M, z ∈ M and yzy = y then yxz ∈ K and zxy ∈ K. More generally, we establish a similar characterization of those directed graphs having edges are labelled with elements of M which have the property that for every such relational morphism there is a choice of related group elements making the corresponding labelled graph "commute". We call these "inevitdbleM-graph". We establish, using this characterization, an effective procedure for deciding from the multiplication table for M whether an "M-graph" is inevitable. A significant stepping–stone towards this was Tilson's 1986 construction which established the Type II Conjecture for regular monoid elements, and this construction is used here in a slightly modified form. But substantial credit should also be given to Henckell, Margolis, Meakin and Rhodes, whose recent independent work follows lines very similar to our own.

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We show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

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ABSTRACT. We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ infinitary formulae and finitely many parameters from 21. This extends the results of [1].To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label. Introduction.We say that a recursive structure 21 is A\\-stable if, for every recursive structure 03 = 21, every isomorphism from 03 to 21 is A° in Kleene's hyperarithmetical hierarchy. We shall show that, under certain assumptions, 21 is A°-stable iff there exists a "formally A° enumeration of 21."The basic outline of our argument is as in [1], where such a result was obtained for finite a. However, certain technicalities prevent the generalization to infinite a from being as straightforward as might be expected. The argument seems most easily described in terms of recursive labelling systems, which were introduced in [1] for a similar purpose.The two basic results for a-systems are obtained in §1 and applied in the succeeding sections to the question of A°-stability.These basic results are also used in the related topic of A°-categoricity [2]. In §1 we define the notion of a recursively a-guided recursive labelling system, or a-system, and in Proposition 1 state the desired result similar to that of [lj involving the existence of r.e. points of 2N and of labellings of A^ paths in an

ABSTRACT. We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ infinitary formulae and finitely many parameters from 21. This extends the results of [1].To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label. Introduction.We say that a recursive structure 21 is A\\-stable if, for every recursive structure 03 = 21, every isomorphism from 03 to 21 is A° in Kleene's hyperarithmetical hierarchy. We shall show that, under certain assumptions, 21 is A°-stable iff there exists a "formally A° enumeration of 21."The basic outline of our argument is as in [1], where such a result was obtained for finite a. However, certain technicalities prevent the generalization to infinite a from being as straightforward as might be expected. The argument seems most easily described in terms of recursive labelling systems, which were introduced in [1] for a similar purpose.The two basic results for a-systems are obtained in §1 and applied in the succeeding sections to the question of A°-stability.These basic results are also used in the related topic of A°-categoricity [2]. In §1 we define the notion of a recursively a-guided recursive labelling system, or a-system, and in Proposition 1 state the desired result similar to that of [lj involving the existence of r.e. points of 2N and of labellings of A^ paths in an

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