Group-like structures
	Totality^α	Associativity	Identity	Divisibility^β	Commutativity
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
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
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Associative quasigroup	Required	Required	Unneeded	Required	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required
^α The closure axiom, used by many sources and defined differently, is equivalent. ^β Here, divisibility refers specifically to the quasigroup axioms.

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.^[1]^[2]

A partial groupoid is a partial algebra.

Partial semigroup

A partial groupoid $(G,\circ )$ is called a partial semigroup if the following associative law holds:^[3]

For all $x,y,z\in G$ such that $x\circ y\in G$ and $y\circ z\in G$ , the following two statements hold:

$x\circ (y\circ z)\in G$ if and only if $(x\circ y)\circ z\in G$ , and
$x\circ (y\circ z)=(x\circ y)\circ z$ if $x\circ (y\circ z)\in G$ (and, because of 1., also $(x\circ y)\circ z\in G$ ).

References