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, one often is challenged by the lack of negation when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient. 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 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 the various theories. Special attention is paid to meta-tangible 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. The systems studied here are "ground" systems, insofar as they are the underlying structure which can be studied via other "module" systems.Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby explaining the mysterious link between classical algebraic results and their tropical and hyperfield analogs. The tropicalization functor indicates analogs of classical algebraic notions, with applications to determinants, linear algebra, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras.