Duality and interval analysis over idempotent semirings

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⊗X⪯B (see [3], ). The purpose of this paper is to consider a dual product, denoted ⊙, and the dual residuation of matrices, in order to solve the following inequality A⊗X⪯X⪯B⊙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 such as they were introduced in [25].