Partial groupoid

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.{{cite book|editor=Ben Silver|title=Nineteen Papers on Algebraic Semigroups|publisher=American Mathematical Soc.|isbn=0-8218-3115-1|author=Evseev, A. E.|chapter=A survey of partial groupoids|year=1988}}{{cite book|editor1=Folkert Müller-Hoissen |editor2=Jean Marcel Pallo |editor3=Jim Stasheff|title=Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift|url=https://archive.org/details/associahedratama00mlle|url-access=limited|year=2012|publisher=Springer Science & Business Media|isbn=978-3-0348-0405-9|pages=[https://archive.org/details/associahedratama00mlle/page/n31 11] and 82}}

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:{{cite journal |last1=Schelp |first1=R. H. |title=A partial semigroup approach to partially ordered sets |journal=Proceedings of the London Mathematical Society |date=1972 |volume=3 |issue=1 |pages=46–58 |doi=10.1112/plms/s3-24.1.46 |url=https://academic.oup.com/plms/article/s3-24/1/46/1572363 |access-date=1 April 2023|url-access=subscription }}

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 $( x \circ y) \circ z \in G$ ).

Partial groupoid

Partial semigroup

References

Further reading