Semigroupoid

Group-like structures
	Totality	Associativity	Identity	Invertibility	Commutativity
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small Category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Inverse Semigroup	Required	Required	Unneeded	Required	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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.

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.