# Supertropical Monoids II: Lifts, Transmissions, and Equalizers

## TL;DR

This paper advances the theory of supertropical monoids and semirings by exploring lifts, transmissions, and equivalence relations, facilitating tangible factorizations and deepening understanding of their algebraic structure.

## Contribution

It introduces explicit classifications of equivalence relations on supertropical monoids and analyzes their properties, addressing challenges from ghost products of tangible elements.

## Key findings

- Constructed and classified equivalence relations on supertropical monoids.
- Analyzed properties of these relations with focus on ghost product issues.
- Enhanced understanding of factorizations in supertropical algebra.

## Abstract

The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are constructed easily, and utilized effectively for supertropical semirings, whereas ideals are too special for semirings. Aiming for tangible factorizations, certain types of such equivalence relations are constructed and classified explicitly in this paper, followed by a profound study of their characteristic properties with special emphasis on difficulties arising from ghost products of tangible elements.