Categories with negation

TL;DR

This paper develops a categorical framework for algebraic structures lacking negation but with a negation-like map, including tropical algebras and hyperfields, and explores their geometric and module-theoretic properties.

Contribution

It extends the theory of $ T$-systems to include ground and module systems, introduces prime systems for geometry, and generalizes module concepts with semi-abelian categories.

Findings

Polynomial systems over prime systems are prime.

Weak Nullstellensatz holds for prime systems.

Dimension of polynomial and Laurent polynomial systems is n.

Abstract

We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is prime, and there is a weak Nullstellensatz. Also, the polynomial $\mathcal A[\la_1, \dots, \la_n]$ and Laurent polynomial systems $\mathcal A[[\la_1, \dots, \la_n]]$ in $n$ commuting indeterminates over a $\tT$-semiring-group system have dimension $n$. For module systems, special attention also is paid to tensor products and $\Hom$. Abelian categories are replaced by "semi-abelian" categories (where $\Hom(A,B)$ is not a group) with a negation morphism.