Partial groupoid explained

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

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

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 , andx \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 ).

Further reading

• Book: E.S. Ljapin. A.E. Evseev. The Theory of Partial Algebraic Operations. 1997. Springer Netherlands. 978-0-7923-4609-8.

Notes and References

1. Book: Ben Silver. Nineteen Papers on Algebraic Semigroups. American Mathematical Soc.. 0-8218-3115-1. Evseev, A. E.. A survey of partial groupoids. 1988.
2. Book: Folkert Müller-Hoissen . Jean Marcel Pallo . Jim Stasheff. Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. limited. 2012. Springer Science & Business Media. 978-3-0348-0405-9. 11 and 82.
3. Schelp . R. H. . A partial semigroup approach to partially ordered sets . Proceedings of the London Mathematical Society . 1972 . 3 . 1 . 46–58 . 10.1112/plms/s3-24.1.46 . 1 April 2023.