STABLE PAIRS

For a finite semigroup S and pseudovariety V, (Y, T) is a V-stable pair of S iff Y ⊆ S, T ≤ S and for any relational morphism R : S ⇝ V with V ∈ V there exists a v ∈ V such that Y ⊆ R-1(v) and T ≤ R-1( Stab (v)). X ≤ S is stable if it is generated by an [Formula: see text]-chain {ai} with aiaj = ai for j < i. Given a relation R : S ⇝ A ∈ A (where A denotes the pseudovariety of aperiodic semigroups) that computes PlA(S), we construct a new relation R∞ : S ⇝ (A(M))# that computes A-stable pairs. This proves the main result of this paper: (Y, T) is an A-stable pair of S iff T ≤ ∪ X for some stableX ≤ PlA(S) and Y ⊆ Y' for some Y' ∈ PlA(S) with Y'x = Y' for all x ∈ X. As a corollary we get that if V is a local pseudovariety of semigroups, then V * A has decidable membership problem.

MODULES

This paper introduces the notion of a module over a graph and defines the wreath product and derived module of a relational morphism in this context.

A NEW PROOF OF THE RHODES TYPE II CONJECTURE

For a given finite monoid M we explicitly construct a finite group G and a relational morphism τ:M→G such that only elements of the type II construct Mc relate to 1 under τ. This provides an elementary and constructive proof of the type II conjecture of John Rhodes. The underlying idea is also used to modify the proof of Ash's celebrated theorem on inevitable graphs. For any finite monoid M and any finite graph Γ a finite group G is constructed which "spoils" all labelings of Γ over M which are not inevitable.

CATEGORIES AS ALGEBRA, II

A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (≺) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism φ : M → N of categories determines an optimal decomposition [Formula: see text] where [Formula: see text] denotes semidirect product and D(φ) is the derived category of φ.The theory of the semidirect product of varieties of categories, V * W, is developed. Associated with each variety V of categories is the collection [Formula: see text] of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form [Formula: see text] are shown to stand in the relationship [Formula: see text] The associativity of the semidirect product of varieties follows from this result.Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that [Formula: see text] for both varieties and pseudovarieties of monoids.

The Kernel Category and Variants of the Concatenation Product

For each variety V of finite monoids, we consider three varieties [Formula: see text] of languages defined in the following way: for each alphabet [Formula: see text] is the Boolean algebra generated by the languages of the form L or L0aL1, where [Formula: see text], a ∈ A and the product L0aL1 is deterministic (resp. codeterministic, bideterministic). We present a description of the corresponding varieties of finite monoids. Such descriptions are done in terms of categories, using the kernel category of a relational morphism. We also give some connections with the positive varieties of languages and with the varieties of finite ordered monoids.

INEVITABLE GRAPHS: A PROOF OF THE TYPE II CONJECTURE AND SOME RELATED DECISION PROCEDURES

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.