nLab monus







Let 𝒞\mathcal{C} be a cartesian monoidal category with a power object functor 𝒫:𝒞𝒞\mathcal{P}:\mathcal{C}\to\mathcal{C}, and let MM be a commutative semigroup object in 𝒞\mathcal{C}. There exists a binary endorelation R=(()())R = \left( (-) \leq (-)\right), R:𝒫(M×M)R\colon \mathcal{P}(M \times M) such that for all a,b:Ma,b\colon M, if there exists a c:Mc\colon M such that a+c=ba + c = b, then (ab):R(a \leq b) \colon R. If \leq is a preorder object, and if for all a,b:Ma,b\colon M, there exists a unique smallest element cc such that (ab+c):R(a \leq b + c) \colon R, then MM is a commutative monoid object with monus, and cc is called the monus of aa and bb.

See also

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