A NEW PROOF OF THE RHODES TYPE II CONJECTURE

Abstract

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.