nLab A-group

For affine groups in linear algebra and geometry, see affine group.

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Definition

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

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

An $\mathcal{A}$-group $G$ is commutative or abelian if $(G, \sim, e, m)$ is a braided monoidal setoid.

 Related concepts

• groupal setoid

• A-set

• tensor product of A-sets

• cartesian product of A-sets

• A-monoid

• 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)