# 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 Unneeded Required Unneeded Unneeded Unneeded
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
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 : 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.

## 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.