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 $R_M$ associated to M, obtained by mapping every positive integer $k$ to the minimal integer $R_M(k)$ such that every word $u$ in $M^*$ of length $R_M(k)$ 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 $R_M$ 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 $R_M$, by proving that $k^{L(M)} \leq R_M(k) \leq (k|M|^4)^{L(M)}$. 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 $n \times n$ Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular $D$-length of $\frac{n^2+n+2}{2}$.