Categories with negation

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.