Kernels in tropical geometry and a Jordan-Holder theorem

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, nu-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield(dagger) F, we pass to the semifield(dagger) F(lambda(1), ..., lambda(n)) of fractions of the polynomial semiring(dagger), for which there already exists a well developed theory of kernels, which are normal convex subgroups of F(lambda(1), ..., lambda(n)); the parallel of the zero set now is the 1-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to nu-kernels (Definition 4.1.4) and 1(nu)-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The.-kernels corresponding to tropical hypersurfaces are the 1(nu)-sets of what we call "corner internal rational functions," and we describe.-kernels corresponding to "usual" tropical geometry as nu-kernels which are "corner-internal" and "regular." This yields an explicit description of tropical affine varieties in terms of various classes of nu-kernels. The literature contains many tropical versions of Hilbert's celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between 1(nu)-sets and a class of nu-kernels of the rational nu-semifield(dagger) called polars, originating from the theory of lattice-ordered groups. When F is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal nu-kernels, intersected with the nu-kernel generated by F. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan-Holder theorem for the relevant class of nu-kernels.