The term "groupoid" has at least three definitions currently in use.
The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on
the operator is that the set
is closed under it (i.e., applying the binary
operator to two elements of a given set 
returns a value
which is itself a member of 
). Associativity, commutativity,
etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty.
The numbers of nonisomorphic groupoids of this
type having 
,
2, ... elements are 1, 10, 3330, 178981952, ... (OEIS
A001329), and the corresponding numbers of
nonisomorphic and nonantiisomorphic groupoids are
1, 7, 1734, 89521056, ... (OEIS A001424). An
associative groupoid is called a semigroup.
The second type of groupoid is, roughly, a category which is "group-like" in the sense that every
morphism (or arrow) is
invertible. More precisely, a small groupoid is a
category 
consisting of a set 
of objects and a set 
of arrows
whereby each arrow 
in 
has an inverse arrow 
(also in 
) subject to the identities 
and 
. Here, 
denotes the source of the arrow 
, 
denotes the target of 
, and 
equals the identity arrow 
of an object 
. This notion of groupoid has become widely applied
throughout modern mathematics and is often seen to generalize many group-theoretic
notions in a number of fields; in particular, examples include the fundamental
groupoid of a manifold 
, as well as more general objects
such as Lie groupoids, holonomy
groupoids, Étale groupoids, etc. (Moerdijk and Mrčun 2003).
The third type of groupoid is an algebraic structure first defined by Brandt (1927) and also known as a virtual group. A groupoid with base
(or "over 
") is a set 
with mappings 
and 
from 
onto 
and a partially defined binary operator 
, satisfying the following four conditions:
1. 
is defined iff 
, and in this case 
and 
.
2. Associativity: if either of 
and 
are defined so is the other and they are equal.
3. For each 
,
there are left- and right-identity elements 
and 
respectively, satisfying 
.
4. Each 
has an inverse 
satisfying 
and 
.
Any group is a groupoid with base a single point.
The most basic example of groupoids with base 
is the pair groupoid, where
, and 
, 
, and with multiplication 
. Any equivalence
relation on 
defines a subgroupoid of the pair groupoid.
A useful way to think of a groupoid of the third type is as a parametrized equivalence relation on 
,
as follows. Given a groupoid over 
, define an equivalence
relation on 
by 
for each 
. This equivalence
relation is "parameterized" because there may be more than one element
in 
which give rise to the same equivalence,
that is, 
and 
such that 
and 
.
The second and third definitions are in fact equivalent.
