An explicit algorithm for normal forms in small overlap monoids

James David Mitchell, Maria Tsalakou*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the C(4) condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in C(4) monoids: the word problem is solvable in O(min{|u|, |v|}) time in the size of the input words u and v; the uniform word problem for ⟨A|R⟩ is solvable in O(N2 min {|u|, |v|}) where N is the sum of the lengths of the words in R; and a normal form for any given word u can be found in O(|u|) time. Although Kambites' algorithm for solving the word problem in C(4) monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice.

In this paper, we present an algorithm for computing normal forms in C(4)
monoids that has time complexity O(|u|2) for input word u, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in O(N min{|u|, |v|}) time.
Original languageEnglish
Pages (from-to)394-433
Number of pages40
JournalJournal of Algebra
Early online date17 May 2023
Publication statusPublished - 15 Sept 2023


  • Monoid
  • Semigroup
  • Normal forms
  • Finite presentation
  • Small overlap
  • Small cancellation


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