Semigroupoid

In mathematics, a semigroupoid is a partial algebra which 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, and 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" o "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" o ("g" o "f") = ("h" o "g") o "f".