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

Let [math]\displaystyle{ x,y,z \in G }[/math] such that [math]\displaystyle{ x\circ y\in G }[/math] and [math]\displaystyle{ y\circ z\in G }[/math], then

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

## References

1. ↑ Evseev, A. E. (1988). "A survey of partial groupoids". in Ben Silver. Nineteen Papers on Algebraic Semigroups. American Mathematical Soc.. ISBN 0-8218-3115-1.
2. ↑ Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. Springer Science & Business Media. 2012. pp. 11 and 82. ISBN 978-3-0348-0405-9. https://archive.org/details/associahedratama00mlle.
3. ↑ Shelp, R. H. (1972). "A Partial Semigroup Approach to Partially Ordered Sets". Proc. London Math. Soc. (1972) s3-24 (1). London Mathematical Soc.. pp. 46–58.

## Further reading

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

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