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 nonempty 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 ?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 ?length of M determines the degree of R_M, by proving that k^L(M) ≤ R_M(k) ≤ (kM⁴)^L(M).
To allow applications of this result, we provide the value of the regular ?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 nondeterministic automata, has a regular ?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 = {44:144:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771801},
ISSN = {18688969},
year = {2021},
volume = {187},
editor = {Bl\"{a}ser, Markus and Monmege, Benjamin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13689},
URN = {urn:nbn:de:0030drops136890},
doi = {10.4230/LIPIcs.STACS.2021.44},
annote = {Keywords: Semigroup, monoid, idempotent, automaton}
}
Keywords: 

Semigroup, monoid, idempotent, automaton 
Collection: 

38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021) 
Issue Date: 

2021 
Date of publication: 

10.03.2021 