Rank preservers of matrices over additively idempotent and multiplicatively cancellative semirings

TL;DR

This paper characterizes linear operators that preserve matrix rank over specific semirings, extending known results from Boolean algebra and max algebra to tropical semirings.

Contribution

It generalizes rank-preserving operator characterizations to additively idempotent and multiplicatively cancellative semirings, including tropical semirings.

Findings

Characterization of rank-preserving linear operators over these semirings

Extension of Boolean algebra and max algebra results

Applicability to max-plus algebra and tropical semirings

Abstract

Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.