Partial groupoid

Group-like structures
	Totality	Associativity	Identity	Invertibility	Commutativity
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small Category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Inverse Semigroup	Required	Required	Unneeded	Required	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.

A partial groupoid is a partial algebra.

Partial semigroup

A partial groupoid [math]\displaystyle{ (G,\circ) }[/math] is called a partial semigroup if the following associative law holds:

For all [math]\displaystyle{ x,y,z \in G }[/math] such that [math]\displaystyle{ x\circ y\in G }[/math] and [math]\displaystyle{ y\circ z\in G }[/math], the following two statements hold:

1. [math]\displaystyle{ x \circ (y \circ z) \in G }[/math] if and only if [math]\displaystyle{ ( x \circ y) \circ z \in G }[/math], and
2. [math]\displaystyle{ x \circ (y \circ z ) = ( x \circ y) \circ z }[/math] if [math]\displaystyle{ x \circ (y \circ z) \in G }[/math] (and, because of 1., also [math]\displaystyle{ ( x \circ y) \circ z \in G }[/math]).

Further reading

• E.S. Ljapin; A.E. Evseev (1997). The Theory of Partial Algebraic Operations. Springer Netherlands. ISBN 978-0-7923-4609-8.