In our study, much like the theory of algebraic geometry, we have
established a connection between tropical geometry and algebra. Namely, we
establish a correspondence between tropical varieties and an algebraic
structures named "kernels" in the semifield of fractions (viewed as a subset of the semifield of functions over the maxplus semifield). The kernels serve as
the analogue to ideals while the semifield of fractions takes the place of
the polynomial ring. We have found a way to represent the corner loci
defined by a tropical polynomial by an equation in the semifield of fractions.
The resulting equation gives rise to a congruence which is encapsulated in the
structure of a kernel (like zero sets and ideals in algebraic geometry). Being
an idempotent semifield, the semifield of fractions is also a "lattice ordered
group" (a group whose elements are endowed with a lattice structure
compatible with the group operation) and the kernels are normal convex
lsubgroups there. The theory of lattice ordered group has been studied
extensively by Birkhoff, Medvedev, Kopytov, Steinberg, Anderson, Feil and
many other mathmaticials. It offers a whole new set of algebraic tools to
study tropical varieties.
