Partial Groupoid

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.

A partial groupoid is a partial algebra.

Partial semigroup

A partial groupoid $(G,\circ)$ is called a partial semigroup (also called semigroupoid, semicategory, naked category, or precategory) if the following associative law holds:

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 $( x \circ y) \circ z \in G$, and
# $x \circ (y \circ z ) = ( x \circ y) \circ z$ if $x \circ (y \circ z) \in G$ (and, because of 1., also $( x \circ y) \circ z \in G$).

References

Further reading

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Algebraic structures

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