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In mathematics, a semigroupoid is a partial algebra[clarification needed] that satisfies the axioms for a small[citation needed] category, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups, 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 morphisms from A to B. If f is in Mor(A,B), we write f : AB.
  • 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 : AB and g : BC is written as gf or gf. (Some authors write it as fg.)

such that the following axiom holds:

  • (associativity) if f : AB, g : BC and h : CD then h ∘ (gf) = (hg) ∘ 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.