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Semigroupoid

Summary

Group-like structures
Total Associative Identity Divisible Commutative
Partial magma Unneeded Unneeded Unneeded Unneeded Unneeded
Semigroupoid Unneeded Required Unneeded Unneeded Unneeded Small category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Commutative groupoid Unneeded Required Required Required Required
Magma Required Unneeded Unneeded Unneeded Unneeded
Commutative magma Required Unneeded Unneeded Unneeded Required
Quasigroup Required Unneeded Unneeded Required Unneeded
Commutative quasigroup Required Unneeded Unneeded Required Required
Unital magma Required Unneeded Required Unneeded Unneeded
Commutative unital magma Required Unneeded Required Unneeded Required
Loop Required Unneeded Required Required Unneeded
Commutative loop Required Unneeded Required Required Required
Semigroup Required Required Unneeded Unneeded Unneeded
Commutative semigroup Required Required Unneeded Unneeded Required
Associative quasigroup Required Required Unneeded Required Unneeded
Commutative-and-associative quasigroup Required Required Unneeded Required Required
Monoid Required Required Required Unneeded Unneeded
Commutative monoid Required Required Required Unneeded Required
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required

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.

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

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  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.