abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a partial

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

(also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. A partial groupoid is a

partial algebra In abstract algebra, a partial algebra is a generalization of universal algebra to partial operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More ...

Partial semigroup

A partial groupoid

(G,\circ)

is called a partial semigroup if the following

associative law In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

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: #

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

x \circ (y \circ z) \in G

(and, because of 1., also

( x \circ y) \circ z \in G

Partial semigroup

References

Further reading