Homomorphisms between the semirings of Boolean functions on 1-dimensional tropical fans

TL;DR

This paper proves a conjecture that all semiring homomorphisms between Boolean function semirings on 1-dimensional tropical fans are geometric, enabling a systematic method to find all such morphisms.

Contribution

It confirms the conjecture that semiring homomorphisms are geometric and provides a new approach to classify morphisms between 1-dimensional tropical fans.

Findings

All semiring homomorphisms are geometric.

A method to classify morphisms between tropical fans.

Validation of the conjecture in the 1-dimensional case.

Abstract

In a previous work, we stated the conjecture that any semiring homomorphism between the Boolean function semirings on 1-dimensional tropical fans has the property called geometric. In this paper, we show that the conjecture is true. As an application, we establish a way to find all the morphisms between given 1-dimensional tropical fans.