Semigroup Forum

, Volume 97, Issue 3, pp 493–522 | Cite as

Gcd-monoids arising from homotopy groupoids

  • Friedrich Wehrung
Research Article


The interval monoid \(\Upsilon ({P})\) of a poset P is defined by generators [xy], where \(x\le y\) in P, and relations \([x,x]=1\), \([x,z]=[x,y]\cdot [y,z]\) for \(x\le y\le z\). It embeds into its universal group \(\Upsilon ^{\pm }({P})\), the interval group of P, which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results:
  • The monoid \(\Upsilon ({P})\) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice).

  • For every group G, there is a connected poset P of height 2 such that \(\Upsilon ({P})\) is a gcd-monoid and G is a free factor of \(\Upsilon ^{\pm }({P})\) by a free group. Moreover, P can be taken to be finite iff G is finitely presented.

  • For every finite poset P, the monoid \(\Upsilon ({P})\) can be embedded into a free monoid.

  • Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid \({\mathrm{U}_\mathrm{mon}}({S})\) of any category S. This enables us, in particular, to characterize the embeddability of \({\mathrm{U}_\mathrm{mon}}({S})\) into a group, by stating that it holds at the hom-set level. We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group.

We illustrate our results by various examples and counterexamples.


Monoid Group Groupoid Category Conical Cancellative Gcd-monoid Simplicial complex Chain complex Barycentric subdivision Universal monoid Universal group Interval monoid Homotopy groupoid Spindle Highlighting expansion 


  1. 1.
    Adjan, S.I.: Defining relations and algorithmic problems for groups and semigroups. Trudy Mat. Inst. Steklov. 85, 123 (1966)MathSciNetGoogle Scholar
  2. 2.
    Bergman, G.M.: On monoids, \(2\)-firs, and semifirs. Semigroup Forum 89(2), 293–335 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brieskorn, E., Saito, K.: Artin-gruppen und Coxeter-gruppen. Invent. Math. 17, 245–271 (1972)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dehornoy, P.: Complete positive group presentations. J. Algebra 268(1), 156–197 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dehornoy, P.: Foundations of Garside Theory, EMS Tracts in Mathematics, vol. 22, European Mathematical Society (EMS), Zürich: with François Digne. Eddy Godelle, Daan Krammer and Jean Michel (2015)Google Scholar
  6. 6.
    Dehornoy, P.: Multifraction reduction I: the 3-Ore case and Artin-Tits groups of type FC. J. Comb. Algebra 1(2), 185–228 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dehornoy, P., Dynnikov, I., Rolfsen, D., Wiest, B.: Why are Braids Orderable? Panoramas et Synthèses [Panoramas and Syntheses], vol. 14. Société Mathématique de France, Paris (2002)zbMATHGoogle Scholar
  8. 8.
    Dehornoy, P., Wehrung, F.: Multifraction reduction III: the case of interval monoids. J. Comb. Algebra 1(4), 341–370 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Freyd, P.: Redei’s finiteness theorem for commutative semigroups. Proc. Am. Math. Soc. 19, 1003 (1968)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Garside, F.A.: The braid group and other groups. Q. J. Math. Oxford Ser. (2) 20, 235–254 (1969)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Higgins, P.J.: Notes on Categories and Groupoids, Van Nostrand Reinhold Co., London-New York-Melbourne, Van Nostrand Reinhold Mathematical Studies, No. 32 (1971)Google Scholar
  12. 12.
    Meakin, J.: Groups and semigroups: connections and contrasts, Groups St. Andrews: Vol. 2, London Math. Soc. Lecture Note Ser., vol. 340, Cambridge University Press, Cambridge 2007, 357–400 (2005)Google Scholar
  13. 13.
    Remmers, J.H.: On the geometry of semigroup presentations. Adv. Math. 36(3), 283–296 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Reynaud, E.: Algebraic fundamental group and simplicial complexes. J. Pure Appl. Algebra 177(2), 203–214 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rotman, J.J.: An Introduction to Algebraic Topology. Graduate Texts in Mathematics, vol. 119. Springer, New York (1988)zbMATHGoogle Scholar
  17. 17.
    Rotman, J.J.: An Introduction to the Theory of Groups. Graduate Texts in Mathematics, vol. 148, 4th edn. Springer, New York (1995)CrossRefGoogle Scholar
  18. 18.
    Spehner, J.-C.: Présentations et présentations simplifiables d’un monoïde simplifiable. Semigroup Forum 14(4), 295–329 (1977)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Spehner, J.-C.: Every finitely generated submonoid of a free monoid has a finite Mal\(^\prime \) cev’s presentation. J. Pure Appl. Algebra 58(3), 279–287 (1989)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LMNO, CNRS UMR 6139 Département de MathématiquesUniversité de Caen NormandieCaen CedexFrance

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