nLab A-monoid

Definition

An $\mathcal{A}$-monoid or affine monoid or antithesis monoid is an $\mathcal{A}$-set $M$ with an $\mathcal{A}$-function $m:M \otimes M \to M$ from the tensor product $M \otimes M$ to $M$ and an element $e$ in $M$ such that $(M, \sim, e, m)$ is a monoidal setoid.

An $\mathcal{A}$-monoid $M$ is strong if $m$ is instead a function from the cartesian product $M \times M$ to $M$.

An $\mathcal{A}$-monoid $M$ is commutative if $(M, \sim, e, m)$ is a braided monoidal setoid.

 Related concepts

• monoidal setoid

• A-set

• tensor product of A-sets

• cartesian product of A-sets

• A-group

• A-ring

 References

• Michael Shulman, Affine logic for constructive mathematics. Bulletin of Symbolic Logic, Volume 28, Issue 3, September 2022. pp. 327 - 386 (doi:10.1017/bsl.2022.28, arXiv:1805.07518)