Semicategory

In mathematics, a semigroupoid (also called semicategory or precategory) is a partial algebra that satisfies the axioms for a smallcategory, 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:

such that the following axiom holds:

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