  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.
