Partial groupoid

Template:Short description Template:Group-like structures 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 $(x\circ y)\circ z\in G$).

Further reading

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