Duality and interval analysis over idempotent semirings

TL;DR

This paper explores duality and interval analysis in idempotent semirings, introducing dual products and residuation to solve complex inequalities and extending these results to interval semirings.

Contribution

It introduces a dual product and dual residuation for matrices over idempotent semirings, providing new methods to solve inequalities and extend results to interval semirings.

Findings

Conditions for the existence of a non-linear projector are established.

Dual residuation techniques enable solving complex inequalities.

Results are extended to semirings of intervals.

Abstract

In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities $A \otimes X \preceq B$. The purpose of this paper is to consider a dual product, denoted $\odot$, and the dual residuation of matrices, in order to solve the following inequality $ A \otimes X \preceq X \preceq B \odot X$. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals.