Partial groupoid

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 ${\displaystyle (x\circ y)\circ z\in G}$).