2026-08-06T13:42:39Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13689 2024-03-06T10:52:40Z ddc:004 open_access

A Ramsey Theorem for Finite Monoids Jecker, Ismaël Semigroup monoid idempotent automaton 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ismaël Jecker 2021 Is Part Of LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2021.44 urn:nbn:de:0030-drops-136890 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.44 eng https://creativecommons.org/licenses/by/4.0/legalcode