Triples and systems: a general algebraic structure theory applicable to tropical mathematics

Our goal is to present an axiomatic algebraic theory which unifies
and “explains” aspects of tropical algebra, hyperfields, and fuzzy rings in terms
of classical algebraic concepts, especially negation, which may not exist a priori.
It was motivated by an attempt to understand whether or not it is coincidental
that basic algebraic theorems are mirrored in supertropical algebra, and was
spurred by the realization that some of the same results are obtained in parallel
research on hyperfields, fuzzy rings, and matroids.
These and many other algebraic theories involve the study of a set T with
incomplete structure that can be understood better by embedding T in a larger
set A endowed with more algebraic structure.
Often the algebra A is a semi-algebra, such as the max-plus algebra, which
does not necessarily have negatives, so much of the research has become a
project of developing a general theory of semi-algebras and their modules,
with suitable axioms that permit one to prove theorems in valuation theory
(tropicalization), linear algebra (joint with Akian and Gaubert), matrix theory
and quadratic forms (joint with Chapman and Niv), geometry, Grassmann
algebras (joint with Gatto), and homology (joint with Jun and Mincheva).
We include a 5-minute introduction to tropical mathematics, to keep the
talk self-contained.