A Ramsey Theorem For Finite Monoids

Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid M, it is natural to ask how long a sequence of elements of M needs to be to ensure the presence of consecutive idempotent factors. This question is formalised through the notion of the Ramsey function RM associated to M, obtained by mapping every k E N to the minimal integer RM(k) such that every word u E M* of length RM(k) contains k consecutive non-empty factors that correspond to the same idempotent element of M. In this work, we study the behaviour of the Ramsey function RM by investigating the regular D -length of M, defined as the largest size L(M) of a submonoid of M isomorphic to the set of natural numbers {1, 2,..., L(M)} equipped with the max operation. We show that the regular D -length of M determines the degree of RM, by proving that kL(m) < RM(k) < (k1m14)L(M) To allow applications of this result, we provide the value of the regular D -length of diverse monoids. In particular, we prove that the full monoid of n x n Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular D-length of '2+2' 2.