Projective systemic modules

TL;DR

This paper extends classical module theory to systems like tropical algebra and hyperrings, establishing foundational results such as the Dual Basis Lemma in a unified framework.

Contribution

It introduces a general theory of projective modules and splitting in systems, unifying tropical algebra, hyperrings, and fuzzy rings.

Findings

Proves a Dual Basis Lemma for systems.

Establishes versions of Schanuel's Lemma in this context.

Provides a unified approach to classical theorems in tropical and hyperring theory.

Abstract

We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.