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

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

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

References

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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