Semigroupoid

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small 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. Semigroupoids have applications in the structural theory of semigroups.

Group-like structures
Totalityα Associativity Identity Divisibilityβ Commutativity
Partial magma UnneededUnneededUnneededUnneededUnneeded
Semigroupoid UnneededRequiredUnneededUnneededUnneeded
Small category UnneededRequiredRequiredUnneededUnneeded
Groupoid UnneededRequiredRequiredRequiredUnneeded
Magma RequiredUnneededUnneededUnneededUnneeded
Quasigroup RequiredUnneededUnneededRequiredUnneeded
Unital magma RequiredUnneededRequiredUnneededUnneeded
Loop RequiredUnneededRequiredRequiredUnneeded
Semigroup RequiredRequiredUnneededUnneededUnneeded
Associative quasigroup RequiredRequiredUnneededRequiredUnneeded
Monoid RequiredRequiredRequiredUnneededUnneeded
Commutative monoid RequiredRequiredRequiredUnneededRequired
Group RequiredRequiredRequiredRequiredUnneeded
Abelian group RequiredRequiredRequiredRequiredRequired
^α The closure axiom, used by many sources and defined differently, is equivalent.
^β Here, divisibility refers specifically to the quasigroup axioms.

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 : 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 ∘ 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 ∘ (g ∘ f) = (h ∘ g) ∘ f.
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