Editing Group (mathematics) (section)

== Generalizations ==
{{Group-like structures}}
More general structures may be defined by relaxing some of the axioms defining a group.{{sfn|Mac Lane|1998}}{{sfn|Denecke|Wismath|2002}}{{sfn|Romanowska|Smith|2002}} The table gives a list of several structures generalizing groups.

For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a [[monoid]]. The [[natural number]]s <math>\mathbb N</math> (including zero) under addition form a monoid, as do the nonzero integers under multiplication {{tmath|1= (\Z \smallsetminus \{0\}, \cdot) }}. Adjoining inverses of all elements of the monoid <math>(\Z \smallsetminus \{0\}, \cdot)</math> produces a group {{tmath|1= (\Q \smallsetminus \{0 \}, \cdot) }}, and likewise adjoining inverses to any (abelian) monoid {{tmath|1= M }} produces a group known as the [[Grothendieck group]] of {{tmath|1= M }}.

A group can be thought of as a [[small category]] with one object {{tmath|1= x }} in which every morphism is an isomorphism: given such a category, the set <math>\operatorname{Hom}(x,x)</math> is a group; conversely, given a group {{tmath|1= G }}, one can build a small category with one object {{tmath|1= x }} in which {{tmath|1= \operatorname{Hom}(x,x) \simeq G }}.
More generally, a [[groupoid]] is any small category in which every morphism is an isomorphism.
In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: {{tmath|1= fg }} is defined only when the source of {{tmath|1= f }} matches the target of {{tmath|1= g }}.
Groupoids arise in topology (for instance, the [[fundamental groupoid]]) and in the theory of [[stack (mathematics)|stacks]].

Finally, it is possible to generalize any of these concepts by replacing the binary operation with an [[arity|{{mvar|n}}-ary]]<!-- Use mvar template instead of LaTeX formatting for n in this section so that the link is colored properly --> operation (i.e., an operation taking {{mvar|n}} arguments, for some nonnegative integer {{mvar|n}}). With the proper generalization of the group axioms, this gives a notion of [[n-ary group|{{mvar|n}}-ary group]].{{sfn|Dudek|2001}} 

{| class="wikitable" style="text-align: center"
|+Examples
|-
 !Set
 ! colspan=2|[[Natural number]]s{{br}}{{tmath|1= \N }}
 ! colspan=2|[[Integer]]s{{br}}{{tmath|1= \Z }}
 ! colspan=4|[[Rational number]]s&nbsp;{{tmath|1= \Q }}{{br}}[[Real number]]s&nbsp;{{tmath|1= \R }}{{br}}[[Complex number]]s&nbsp;{{tmath|1= \C }}
 ! colspan=2|[[Integers modulo n|Integers modulo 3]]{{br}}{{tmath|1= \Z / n\Z = \{0, 1, 2\} }}
|-
 !Operation
 ! +
 ! ×
 ! +
 ! ×
 ! +
 ! −
 ! ×
 ! ÷
 ! +
 ! ×
|-
 ![[Total function|Total]]
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{yes|yes}}
|-
 !Identity
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{yes|yes}}
|-
 !Inverse
 | {{no|no}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | only if {{tmath|1= a \ne 0 }}
 | {{no|no}}
 | {{yes|yes}}
 | only if {{tmath|1= a \ne 0}}
|-
 !Divisibility
 | {{no|no}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{yes|yes}}
 | only if {{tmath|1= a \ne 0}}
 | only if {{tmath|1= a \ne 0}}
 | {{yes|yes}}
 | {{no|no}}
|-
 !Associative
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{yes|yes}}
|-
 !Commutative
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{no|no}}
 | {{yes|yes}}
 | {{yes|yes}}
|-
 !Structure
 | [[monoid]] || [[monoid]]
 | [[abelian group]]
 | [[monoid]]
 | [[abelian group]]
 | [[quasigroup]]
 | [[monoid]]
 | [[quasigroup]]{{br}}(with 0 removed)
 | [[abelian group]]
 | [[monoid]]
|-
|}

{{clear}}