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 , ∘ ) {\displaystyle (G,\circ )} is called a partial semigroup if the following associative law holds: For all x , y , z ∈ G {\displaystyle x,y,z\in G} such that x ∘ y ∈ G {\displaystyle x\circ y\in G} and y ∘ z ∈ G {\displaystyle y\circ z\in G} , the following two statements hold:
x ∘ ( y ∘ z ) ∈ G {\displaystyle x\circ (y\circ z)\in G} if and only if ( x ∘ y ) ∘ z ∈ G {\displaystyle (x\circ y)\circ z\in G} , and
x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z {\displaystyle x\circ (y\circ z)=(x\circ y)\circ z} if x ∘ ( y ∘ z ) ∈ G {\displaystyle x\circ (y\circ z)\in G} (and, because of 1., also ( x ∘ y ) ∘ z ∈ G {\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.