Linear isomorphisms preserving Green's relations for matrices over 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 ${\le }_{\mathcal{L}},{\le }_{\mathcal{R}},{\le }_{\mathcal{J}}$. 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.

14 pages

References in corpus (2)

• Tropical matrix duality and Green's D relation
• Exact rings and semirings