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Difference between revisions of "Groupoid"

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A [[universal algebra]] with one [[binary operation]]. It is the broadest class of such algebras: groups, semi-groups, quasi-groups — all these are groupoids of a special type. An important concept in the theory of groupoids is that of isotopy of operations. On a set $G$ let there be defined two binary operations, denoted by $(\cdot)$ and $(\circ)$; they are isotopic if there exist three one-to-one mappings $\alpha$, $\beta$ and $\gamma$ of $G$ onto itself such that $a\cdot b=\gamma^{-1}(\alpha a\circ\beta b)$ for all $a,b\in G$ (cf. [[Isotopy (in algebra)]]). A groupoid that is isotopic to a [[Quasi-group|quasi-group]] is itself a quasi-group; a groupoid with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.
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A term introduced by H. Brandt [[#References|[a1]]]. A groupoid may conveniently be defined as a (small) [[Category|category]] in which every morphism is an isomorphism; equivalently, it is a set $G$ equipped with a unary operation $g\mapsto g^{-1}$ and a partial binary operation $(g,h)\mapsto gh$ satisfying
 
 
A groupoid with cancellation is a groupoid in which either of the equations $ab=ac$, $ba=ca$ implies $b=c$, where $a$, $b$ and $c$ are elements of the groupoid. Any groupoid with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a groupoid with division, that is, a groupoid in which the equations $ax=b$ and $ya=b$ are solvable (but do not necessarily have unique solutions).
 
 
 
A set with one partial binary operation (i.e. one not defined for all pairs of elements) is said to be a partial groupoid. Any partial subgroupoid of a free partial groupoid is free.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Boruvka,  "Foundations of the theory of groupoids and groups" , Wiley  (1976)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" , Springer  (1958)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
There is another, conflicting, use of the term "groupoid" in mathematics, which was introduced by H. Brandt [[#References|[a1]]]. A groupoid may conveniently be defined as a (small) [[Category|category]] in which every morphism is an isomorphism; equivalently, it is a set $G$ equipped with a unary operation $g\mapsto g^{-1}$ and a partial binary operation $(g,h)\mapsto gh$ satisfying
 
  
 
1) $gg^{-1}$ and $g^{-1}g$ are always defined;
 
1) $gg^{-1}$ and $g^{-1}g$ are always defined;

Revision as of 17:59, 13 December 2015


A term introduced by H. Brandt [a1]. A groupoid may conveniently be defined as a (small) category in which every morphism is an isomorphism; equivalently, it is a set $G$ equipped with a unary operation $g\mapsto g^{-1}$ and a partial binary operation $(g,h)\mapsto gh$ satisfying

1) $gg^{-1}$ and $g^{-1}g$ are always defined;

2) $gh$ is defined if and only if $g^{-1}g=hh^{-1}$;

3) if $gh$ and $hk$ are defined, then $(gh)k$ and $g(hk)$ are defined and equal;

4) each of $g^{-1}gh$, $hg^{-1}g$, $gg^{-1}h$, and $hgg^{-1}$ is equal to $h$ if it is defined.

Groupoids, as a special case of categories, play an important role in many areas of application of category theory, including algebra [a2], different geometry [a3] and topology [a4], [a5].

References

[a1] H. Brandt, "Ueber eine Verallgemeinerung des Gruppenbegriffes" Math. Ann. , 96 (1926) pp. 360–366
[a2] P.J. Higgins, "Categories and groupoids" , v. Nostrand-Reinhold (1971)
[a3] Ch. Ehresmann, "Structures locales et catégories ordonnés" , Oeuvres complètes et commentées , Supplément aux Cahiers de Topologie et Géométrie Différentielle Catégoriques , Partie II (1980)
[a4] R. Brown, "Elements of modern topology" , McGraw-Hill (1968)
[a5] R. Brown, "From groups to groupoids: a brief survey" Bull. London Math. Soc. , 19 (1987) pp. 113–134
How to Cite This Entry:
Groupoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Groupoid&oldid=36915
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article