  Our objective in this project is three-fold, the first two covered in this
paper. In tropical mathematics, as well as other mathematical theories
involving semirings, when trying to formulate the tropical versions of
classical algebraic concepts for which the negative is a crucial ingredient,
such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and
Poisson algebras, one often is challenged by the lack of negation. Following an
idea originating in work of Gaubert and the Max-Plus group and brought to
fruition by Akian, Gaubert, and Guterman, we study algebraic structures with
negation maps, called \textbf{systems}, in the context of universal algebra,
showing how these unify the more viable (super)tropical versions, as well as
hypergroup theory and fuzzy rings, thereby "explaining" similarities in their
theories. Special attention is paid to \textbf{meta-tangible} $\mathcal
T$-systems, whose algebraic theory includes all the main tropical examples and
many others, but is rich enough to facilitate computations and provide a host
of structural results.
  Basic results also are obtained in linear algebra, linking determinants to
linear independence.
  Formulating the structure categorically enables us to view the
tropicalization functor as a morphism, thereby further explaining the
mysterious link between classical algebraic results and their tropical analogs,
as well as with hyperfields. We utilize the tropicalization functor to propose
tropical analogs of classical algebraic notions.
  The systems studied here might be called "fundamental," since they are the
underlying structure which can be studied via other "module" systems, which is
to be the third stage of this project, involving a theory of sheaves and
schemes and derived categories with a negation map.
