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In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.cite-ref-silver-1-0[1]cite-ref-m-ller-hoissenpallo2012-2-0[2]

A partial groupoid is a partial algebra.

• Partial semigroup

• References

• Further reading

A partial groupoid ( G , ∘ ∘ ) {\displaystyle (G,\circ )} is called a partial semigroup if the following associative law holds:cite-ref-schelp-3-0[3]

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

1. x ∘ ∘ ( y ∘ ∘ z ) ∈ ∈ G {\displaystyle x\circ (y\circ z)\in G} if and only if ( x ∘ ∘ y ) ∘ ∘ z ∈ ∈ G {\displaystyle (x\circ y)\circ z\in G} , and

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

cite-note-silver-11. ↑ citerefevseev-a-e-1988Evseev, 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.

cite-note-m-ller-hoissenpallo2012-22. ↑ citereffolkert-m-ller-hoissenjean-marcel-pallojim-stasheff2012Folkert 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.

cite-note-schelp-33. ↑ citerefschelp1972Schelp, 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.

• citerefe-s-ljapina-e-evseev1997E.S. Ljapin; A.E. Evseev (1997). The Theory of Partial Algebraic Operations. Springer Netherlands. ISBN 978-0-7923-4609-8.