abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

, 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 func ...

(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 (also called semigroupoid, semicategory, naked category, or precategory) if the following

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

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