Linear isomorphisms preserving Green's relations for matrices over semirings

TL;DR

This paper characterizes linear bijections on matrix monoids over anti-negative semifields that preserve Green's relations, with applications to tropical and boolean semirings, enhancing understanding of their algebraic structure.

Contribution

It provides a complete characterization of linear isomorphisms preserving Green's relations over matrices on anti-negative semifields, including tropical and boolean cases.

Findings

Characterization of linear bijections preserving Green's relations.

Results applicable to tropical and boolean semirings.

Additional results for the H relation in specific semirings.

Abstract

In this paper we characterize those linear bijective maps on the monoid of all $n \times n$ square matrices over an anti-negative semifield which preserve and strongly preserve each of Green's equivalence relations $\mathcal{L}, \mathcal{R}, \mathcal{D}, \mathcal{J}$ and the corresponding three pre-orderings $\leq_\mathcal{L}, \leq_\mathcal{R}, \leq_\mathcal{J}$. These results apply in particular to the tropical and boolean semirings, and for these two semirings we also obtain corresponding results for the $\mathcal{H}$ relation.