Projective modules over polyhedral semirings

TL;DR

This paper classifies projective modules over certain idempotent semirings, including those of convex, piecewise-affine functions on polyhedra, linking algebraic structures to geometric objects like weight polyhedra.

Contribution

It extends the classification of projective modules to semirings of convex functions on polyhedra, connecting algebraic and geometric frameworks.

Findings

Classified projective modules over idempotent semirings free on a monoid.

Extended classification to semirings of convex, piecewise-affine functions.

Linked projective modules to convex families of weight polyhedra for GL(n).

Abstract

I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group.