In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a semigroupoid is a partial algebra which satisfies the axioms for a small category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

s in the same way that small categories generalise monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

s and groupoid

Groupoid

In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...

s generalise groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, and have applications in the structural theory of semigroups.

Formally, a semigroupoid consists of:

• a set of things called objects.
• for every two objects A and B a set Mor(A,B) of things called morphism
  Morphism
  In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
  
  s from A to B. If f is in Mor(A,B), we write f : A → B.
• for every three objects A, B and C a binary operation Mor(A,B) × Mor(B,C) → Mor(A,C) called composition of morphisms. The composition of f : A → B and g : B → C is written as g o f or gf. (Some authors write it as fg.)

such that the following axiom holds:

• (associativity) if f : A → B, g : B → C and h : C → D then h o (g o f) = (h o g) o f.

Although the axioms defining a single semigroup are almost identical to those defining a category, one frequently assumes many additional facts about the relationship between categories (functors between them exist, natural transformations between these functors exist, the functors themselves form a category, and so on). The term "semigroup" does not imply any of these assumptions.