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