Partial groupoid

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 ${\displaystyle (M,\circ )}$ is called a partial semigroup if the following associative law holds:^[3]

Let ${\displaystyle x,y,z\in G}$ such that ${\displaystyle x\circ y\in G}$ and ${\displaystyle y\circ z\in G}$, then

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

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. Folkert 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.
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.