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URN: urn:nbn:de:0030-drops-136890
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### A Ramsey Theorem for Finite Monoids

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### Abstract

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 k ∈ ℕ to the minimal integer R_M(k) such that every word u ∈ 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) ≤ R_M(k) ≤ (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 × n Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular 𝒟-length of (n²+n+2)/2.

### BibTeX - Entry

```@InProceedings{jecker:LIPIcs.STACS.2021.44,
author =	{Jecker, Isma\"{e}l},
title =	{{A Ramsey Theorem for Finite Monoids}},
booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
pages =	{44:1--44:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-180-1},
ISSN =	{1868-8969},
year =	{2021},
volume =	{187},
editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
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