nLab A-ring

Definition

An $\mathcal{A}$-ring or affine ring or antithesis ring is an $\mathcal{A}$-set $R$ with an $\mathcal{A}$-function $-:R \to R$, an $\mathcal{A}$-function $(-)+(-):R \otimes R \to R$ from the tensor product $R \otimes R$ to $R$, an element $0$ in $R$, an $\mathcal{A}$-function $(-)\cdot(-):R \otimes R \to R$ from the tensor product $R \otimes R$ to $R$ and an element $1$ in $R$ such that $(R, \sim, 0, +, -, 1, \cdot)$ is a ring setoid.

An $\mathcal{A}$-ring $R$ is

• additively strong if $(-)+(-)$ is instead a function from the cartesian product $R \times R$ to $R$.

• multiplicatively strong if $(-)\cdot(-)$ is instead a function from the cartesian product $R \times R$ to $R$.

• strong if $R$ is both additively strong and multiplicatively strong.

An $\mathcal{A}$-ring $R$ is commutative if $(R, \sim, 1, \cdot)$ is a braided monoidal setoid.

 Related concepts

• ring setoid

• A-set

• tensor product of A-sets

• cartesian product of A-sets

• A-monoid

• A-group

 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)