Partial groupoid

Group-like structures
	Total	Associative	Identity	Divisible
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded
Groupoid	Unneeded	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required
Unital magma	Required	Unneeded	Required	Unneeded
Loop	Required	Unneeded	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded
Associative quasigroup	Required	Required	Unneeded	Required
Monoid	Required	Required	Required	Unneeded
Group	Required	Required	Required	Required

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.¹²

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

Evseev, A. E. (1988). "A survey of partial groupoids". In Ben Silver (ed.). Nineteen Papers on Algebraic Semigroups. American Mathematical Soc. ISBN 0-8218-3115-1.
Folkert Müller-Hoissen; Jean Marcel Pallo; Jim Stasheff, eds. (2012). Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. Springer Science & Business Media. pp. 11 and 82. ISBN 978-3-0348-0405-9.
Schelp, R. H. (1972). "A partial semigroup approach to partially ordered sets". Proceedings of the London Mathematical Society. 3 (1): 46–58. doi:10.1112/plms/s3-24.1.46. Retrieved 1 April 2023.