# Partial groupoid

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{{Short description|Set endowed with a partial binary operation}}
{{Group-like structures}}
In [abstract algebra](/source/abstract_algebra), a '''partial [groupoid](/source/magma_(algebra))''' (also called '''halfgroupoid''', '''pargoid''', or '''partial magma''') is a set endowed with a [partial binary operation](/source/Binary_operation).<ref name="Silver">{{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}}</ref><ref name="Müller-HoissenPallo2012">{{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}}</ref>

A partial groupoid is a [partial algebra](/source/partial_algebra).

== Partial semigroup ==

A partial groupoid <math>(G,\circ)</math> is called a '''partial semigroup''' if the following [associative law](/source/associative_law) holds:<ref name="Schelp">{{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 }}</ref>

For all <math>x,y,z \in G</math> such that <math> x\circ y\in G</math> and <math> y\circ z\in G</math>, the following two statements hold:
# <math>x \circ (y \circ z) \in G</math> if and only if <math>( x \circ y) \circ z \in G</math>, and
# <math>x \circ (y \circ z ) = ( x \circ y) \circ z</math> if <math>x \circ (y \circ z) \in G</math> (and, because of 1., also <math>( x \circ y) \circ z \in G</math>).

== References ==
{{reflist}}

== Further reading ==
* {{cite book|author1=E.S. Ljapin|author2=A.E. Evseev|title=The Theory of Partial Algebraic Operations|year=1997|publisher=Springer Netherlands|isbn=978-0-7923-4609-8}}

Category:Algebraic structures

{{algebra-stub}}

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