Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

P. J. Cameron, James East, D. Fitzgerald, J. D. Mitchell, L. Pebody, T. Quinn-Gregson

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1 Citation (Scopus)

Abstract

For a positive integer n, the full transformation semigroup Tn consists of all self maps of the set {1, …, n} under composition. Any finite semigroup S embeds in some Tn, and the least such n is called the (minimum transformation) degree of S and denoted µ(S). We find degrees for various classes of finite semigroups, including rectangular bands, rectangular groups and null semigroups. The formulae we give involve natural parameters associated to integer compositions. Our results on rectangular bands answer a question of Easdown from 1992, and our approach utilises some results of independent interest con-cerning partitions/colourings of hypergraphs. As an application, we prove some results on the degree of a variant Tan. (The variant Sa = (S, ⋆) of a semigroup S, with respect to a fixed element a ∈ S, has underlying set S and operation x ⋆ y = xay.) It has been previously shown that (Formula presented) if the sandwich element a has rank r, and the upper bound of 2n − r is known to be sharp if r ⩾ n − 1. Here we show that µ(Tna) = 2n − r for r ⩾ n − 6. In stark contrast to this, when r = 1, and the above inequality says (Formula presented), we show that (Formula presented). Among other results, we also classify the 3-nilpotent subsemigroups of Tn, and calculate the maximum size of such a subsemigroup.

Original languageEnglish
Article number16
Number of pages48
JournalCombinatorial Theory
Volume3
Issue number3
Publication statusPublished - 2023

Bibliographical note

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Open Access - Access Right Statement

Copyright 2023 by the author(s).This work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/

Keywords

  • nilpotent semigroup
  • rectangular band
  • hypergraph
  • Transformation semigroup
  • transformation representation
  • semigroup variant

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