Semigroupoid

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small[1][2][3] 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 Invertibility Commutativity
Semigroupoid UnneededRequiredUnneededUnneededUnneeded
Small Category UnneededRequiredRequiredUnneededUnneeded
Groupoid UnneededRequiredRequiredRequiredUnneeded
Magma RequiredUnneededUnneededUnneededUnneeded
Quasigroup RequiredUnneededUnneededRequiredUnneeded
Unital Magma RequiredUnneededRequiredUnneededUnneeded
Loop RequiredUnneededRequiredRequiredUnneeded
Semigroup RequiredRequiredUnneededUnneededUnneeded
Inverse Semigroup RequiredRequiredUnneededRequiredUnneeded
Monoid RequiredRequiredRequiredUnneededUnneeded
Commutative monoid RequiredRequiredRequiredUnneededRequired
Group RequiredRequiredRequiredRequiredUnneeded
Abelian group RequiredRequiredRequiredRequiredRequired
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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.

References
  1. Tilson, Bret (1987). "Categories as algebra: an essential ingredient in the theory of monoids". J. Pure Appl. Algebra. 48 (1–2): 83–198. doi:10.1016/0022-4049(87)90108-3., Appendix B
  2. Rhodes, John; Steinberg, Ben (2009), The q-Theory of Finite Semigroups, Springer, p. 26, ISBN 9780387097817
  3. See e.g. Gomes, Gracinda M. S. (2002), Semigroups, Algorithms, Automata and Languages, World Scientific, p. 41, ISBN 9789812776884, which requires the objects of a semigroupoid to form a set.


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