A note on Garside monoids and $${\mathscr {M}}$$-braces

Research output: Contribution to journalArticlepeer-review


We define an algebraic structure similar to that of a semiring, but without some of the requirements. As it is somehow also similar to the structure of left brace, we call it an M-brace. We present a connection between Garside monoids and more generally lcm-monoids with this algebraic structure. An lcm-monoid M is a left-cancellative monoid such that 1 is the unique invertible element in M, and every pair of elements in M admit an lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids.

Original languageEnglish
Pages (from-to)553-568
Number of pages16
JournalSemigroup Forum
Issue number3
StatePublished - Jun 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Garside monoids
  • Gaussian monoids
  • braces
  • lcm monoids

ASJC Scopus subject areas

  • Algebra and Number Theory


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