Supertropical Monoids III: Factorization and splitting covers

TL;DR

This paper develops a new factorization theory for supertropical monoids using fiber contractions, introducing covers and splitting covers, and explores their quotient structures within a categorical framework.

Contribution

It introduces a novel approach to factorization in supertropical monoids via fiber contractions and covers, expanding the categorical understanding of quotients and splitting covers.

Findings

Tangible factorization into irreducibles achieved

Fiber contractions define new quotient structures

Covers and splitting covers characterized within the framework

Abstract

The category $STROP_m$ of supertropical monoids, whose morphisms are transmissions, has the full--reflective subcategory $STROP$ of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.