A general algebraic structure theory for tropical mathematics

The goal of this overview is to present an axiomatic algebraic theory which unifies, simplifies, and ``explains'' aspects of idempotent algebra, tropical algebra, hyperfields, and fuzzy rings, terms of familiar classical algebraic concepts. It was motivated by an attempt to understand whether or not it is coincidental that basic algebraic theorems are mirrored in supertropical algebra (in work with Izhakian and Knebusch), and was spurred by the realization that some of the same results have been obtained in parallel research on hyperfields and fuzzy rings. Our objective is to hone in on the precise axioms that include these various examples, formulate the axiomatic structure, and describe its uses for linear algebra, exterior algebras, geometry, homology. Collaborators in this project so far include Akian, Gatto, Gaubert, Jun, and Mincheva.