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 max-plus 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
l-subgroups 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.
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