Partial groupoid

Partial semigroup

A partial groupoid ( M , ∘ ) is called a partial semigroup if the following associative law holds:

Let x , y , z ∈ G such that x ∘ y ∈ G and y ∘ z ∈ G , then

1. x ∘ ( y ∘ z ) ∈ G if and only if ( x ∘ y ) ∘ z ∈ G
2. and x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z if x ∘ ( y ∘ z ) ∈ G (and, because of 1., also ( x ∘ y ) ∘ z ∈ G ).