Editing Groupoid (section)

=== Algebraic ===

A groupoid can be viewed as an algebraic structure  consisting of a set with a binary [[partial function]] {{Citation needed|reason=appears to contradict prominent sources such as MathWorld|date=July 2024}}.
Precisely, it is a non-empty set <math>G</math> with a [[unary operation]] {{tmath|1= {}^{-1} : G\to G }}, and a [[partial function]] {{tmath|1= *:G\times G \rightharpoonup G }}.  Here <math>*</math> is not a [[binary operation]] because it is not necessarily defined for all pairs of elements of {{tmath|1= G }}.  The precise conditions under which <math>*</math> is defined are not articulated here and vary by situation.

The operations <math>\ast</math> and <sup>−1</sup> have the following axiomatic properties:  For all {{tmath|1= a }}, {{tmath|1= b }}, and <math>c</math> in {{tmath|1= G }},
# ''[[Associativity]]'': If <math>a*b</math> and <math>b*c</math> are defined, then <math>(a * b) * c</math> and <math>a * (b * c)</math> are defined and are equal.  Conversely, if one of <math>(a * b) * c</math> or <math>a * (b * c)</math> is defined, then they are both defined (and they are equal to each other), and <math>a*b</math> and <math>b * c</math> are also defined.
# ''[[Multiplicative inverse|Inverse]]'': <math>a^{-1} * a</math> and <math>a*{a^{-1}}</math> are always defined.
# ''[[Identity element|Identity]]'': If <math>a * b</math> is defined, then {{tmath|1= a * b * {b^{-1} } = a }}, and {{tmath|1= {a^{-1} } * a * b = b }}.  (The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:
* {{tmath|1= (a^{-1} )^{-1} = a }},
* If <math>a * b</math> is defined, then {{tmath|1= (a * b)^{-1} = b^{-1} * a^{-1} }}.<ref>

Proof of first property: from 2. and 3. we obtain ''a''<sup>−1</sup> = ''a''<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> and (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * (''a''<sup>−1</sup>)<sup>−1</sup>. Substituting the first into the second and applying 3. two more times yields (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> * (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * ''a'' = ''a''.  ✓ <br />
Proof of second property: since ''a'' * ''b'' is defined, so is (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b''.  Therefore (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b'' * ''b''<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' is also defined.  Moreover since ''a'' * ''b'' is defined, so is ''a'' * ''b'' * ''b''<sup>−1</sup> = ''a''.  Therefore ''a'' * ''b'' * ''b''<sup>−1</sup> * ''a''<sup>−1</sup> is also defined.  From 3. we obtain (''a'' * ''b'')<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b'' * ''b''<sup>−1</sup> * ''a''<sup>−1</sup> = ''b''<sup>−1</sup> * ''a''<sup>−1</sup>.  ✓</ref>