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

Abstract

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.