Many algebraic theories involve the study of a set T with fragmented structure which
can be understood better by embedding T in a larger set A endowed with more struc-
ture. Classical examples include the homogeneous components of a graded algebra.
In the direction of tropical mathematics, the max-plus algebra and related tropical
structures were embedded by Izhakian into semirings which are more manageable,
and the same can be said for hypergroups and fuzzy rings.
On the other hand, in mathematical theories involving semirings, one often is chal-
lenged by the lack of negation when trying to formulate the tropical versions of clas-
sical algebraic concepts for which the negative is a crucial ingredient. Developing an
idea of Gaubert in his doctoral dissertation and brought to fruition by Akian, Gau-
bert, and Guterman, we study triples (A, T,(−)) with negation maps, in the context
of universal algebra, showing how these unify the more viable (super)tropical ver-
sions, as well as hypergroup theory and fuzzy rings, thereby “explaining” similarities
in the various theories. Special attention is paid to meta-tangible triples, defined by
the property that a + b ∈ T for all a, b ∈ T for which b /= (−)a.
Furthermore, equality on T generalizes to a relation on A which plays a key
structural role, yielding a system. Their 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. Systems can be “fundamental,” insofar as they provide
the underlying structure, which then is studied via classical structure theory, as well
as linear algebra (in ongoing research with Akian and Gaubert) and through repre-
sentation theory via “module” systems (in ongoing work with Jun and Mincheva,
paralleling research of Connes and Consani).
This approach enables one to view the tropicalization functor as a morphism, the-
reby indicating tropical analogs of such classical algebraic structures as Grassmann
algebras, Lie algebras, Lie superalgebras, Poisson algebras, and Hopf algebras.