Partial groupoid

Group-like structures

	Total	Associative	Identity	Divisible	Commutative
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
Commutative groupoid	Unneeded	Required	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutative magma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Commutative quasigroup	Required	Unneeded	Unneeded	Required	Required
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Commutative unital magma	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded
Commutative loop	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Commutative semigroup	Required	Required	Unneeded	Unneeded	Required
Associative quasigroup	Required	Required	Unneeded	Required	Unneeded
Commutative-and-associative quasigroup	Required	Required	Unneeded	Required	Required
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

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

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

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

References

1. 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.
2. 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.
3. 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.

Further reading

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