Fundamental Theorem of Projective Geometry over Semirings

TL;DR

This paper extends the fundamental theorem of projective geometry to semimodules over semirings, exploring linear algebra and tropical geometry implications, especially for tropical lines.

Contribution

It introduces a version of the fundamental theorem of projective geometry for semimodules over semirings, connecting it with tropical geometry and bases in semimodules.

Findings

Established the fundamental theorem for semimodules over semirings.

Provided explicit results on maps between tropical lines.

Explored geometric connections to tropical geometry.

Abstract

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry