# nLab monus

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

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