A Ramsey Theorem for Finite Monoids

Author Ismaël Jecker

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Ismaël Jecker
  • Institute of Science and Technology, Klosterneuburg, Austria


I wish to thank Michaël Cadilhac, Emmanuel Filiot and Charles Paperman for their valuable insights concerning Green’s relations.

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Ismaël Jecker. A Ramsey Theorem for Finite Monoids. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic language theory
  • Semigroup
  • monoid
  • idempotent
  • automaton


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