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 formula_1 is called a partial semigroup if the following associative law holds:
Let formula_2 such that formula_3 and formula_4, then 