Partial groupoid

In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.[1][2]

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:[3]

Let $x,y,z\in G$ such that $x\circ y\in G$ and $y\circ z\in G$ , then

$x\circ (y\circ z)\in G$ if and only if $(x\circ y)\circ z\in G$
and $x\circ (y\circ z)=(x\circ y)\circ z$ if $x\circ (y\circ z)\in G$ (and, because of 1., also $(x\circ y)\circ z\in G$ ).

Evseev, A. E. (1988). "A survey of partial groupoids". In Ben Silver (ed.). Nineteen Papers on Algebraic Semigroups. American Mathematical Soc. ISBN 0-8218-3115-1.
Folkert Müller-Hoissen; Jean Marcel Pallo; Jim Stasheff, eds. (2012). Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift. Springer Science & Business Media. pp. 11 and 82. ISBN 978-3-0348-0405-9.
Shelp, R. H. (1972). "A Partial Semigroup Approach to Partially Ordered Sets". Proc. London Math. Soc. (1972) s3-24 (1). London Mathematical Soc. pp. 46–58.