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By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{1}=1$, $\underline{x \cdot y}=\underline{x} \cdot \underline{y}$. Notice that the elements of this group have the form $\underline{x_1} \cdot \underline{x_2}^{-1} \cdot \underline{x_3} \cdot \underline{x_4}^{-1} \cdot \dotsc$.

Question 1: Does this group have a name? It is analogous to the Grothendieck group, which is the left adjoint of the forgetful functor from abelian groups to commutative monoids, so perhaps we may call it the non-abelian Grothendieck group?

Question 2: Is there any criterion when an element of this group vanishes?

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    $\begingroup$ I think semigroups theorists most often say maximal group image. Universal group and group completion are fairly common. The second volume of Clifford and Preston gives a detailed account of Malcev and Lambek results. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 7, 2015 at 20:30
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    $\begingroup$ For an alternative approach to the terminology, I can't help noticing that this is the 1-object version of localizing a category at all its morphisms. Since this localization corresponds (correct me if I'm wrong) to taking the 1-truncation of the classifying space of the category, there might be some topological terminology that would be suitable... $\endgroup$
    – Tim Campion
    Commented Jan 7, 2015 at 20:50
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    $\begingroup$ It is the fundamental group of the classifying space of M. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 7, 2015 at 20:56
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    $\begingroup$ @TimCampion, I know of finitely presented inverse monoids with decidable word problem whose universal group has undecidable word problem. I don't think they are finitely presented in the category of monoids but in any event such finitely presented examples must exist. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 8, 2015 at 2:52
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    $\begingroup$ This group is also called the group completion of the monoid. $\endgroup$
    – Omar Antolín-Camarena
    Commented Jan 8, 2015 at 19:49

3 Answers 3

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George Bergman refers to it as the universal enveloping group of the monoid. See Section 3.11, page 81, of his universal algebra notes/forthcoming book: http://math.berkeley.edu/~gbergman/245/

He also references P.M. Cohn's Universal Algebra book, and two papers of Mal'cev in which he establishes conditions for the universal map to be an embedding.

P.S. As far as I know, the construction predates Grothendieck's work; but because of the applications the latter found for the abelian case, his work is better known. In fact, this is not the first time I've seen the general case described as "the nonabelian version of the Grothendieck group"...

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I have seen "(universal) enveloping group of the monoid" used for this construction. Mal'cev has found necessary and sufficient conditions for injectivity of the comparison map.

Anatoliy I.Mal’cev, Über die Einbettung von assoziativen Systemen in Gruppen, Mat. Sb. N.S. 6 (1939) 331-336.

Anatoliy I. Mal’cev, Über die Einbettung von assoziativen Systemen in Gruppen, II, Mat. Sb. N.S. 8 (1940) 251-264.

See also Chapter VII in P. M. Cohn, Universal Algebra, second edition, Reidel, 1981.

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    $\begingroup$ It should be added that the quasivariety of monoids embeddable in a group has no finite basis of quasi-identities and there is no algorithm to decide if a finitely presented monoid embeds in a group. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 7, 2015 at 20:28
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    $\begingroup$ Thanks, the result about the finite basis is in the second paper of Mal'cev. $\endgroup$
    – Andreas Thom
    Commented Jan 7, 2015 at 20:29
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I'd just like to point out that in the many-object case this has been called the fundamental groupoid of a category by Paré, which would suggest that it could be called the fundamental group of a monoid, a possibility strengthened by the fact that, as Benjamin Steinberg points out in the comments, it's the fundamental group of the classifying space of the monoid. Another possibility might be classifying group, but that's probably ill-advised since it's not clear (to me) what it classifies.

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  • $\begingroup$ But if I had to vote, in the single-object case I'd go with group completion as in Omar Antolín-Camarena's comment above. Even if "universal enveloping group" is more standard in the general algebra literature, it just seems overly heavy-handed to me. $\endgroup$
    – Tim Campion
    Commented Jan 8, 2015 at 23:52

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