On conditions for mappings to preserve optimal solutions of semiring-induced valuation algebras

The valuation algebra is a generic algebraic structure which links up with local computation and inference. A projection problem in a valuation algebra concerns the focusing of a collection of information on a local domain. In this paper, according to a classification of variable sets, we present some corresponding necessary conditions for mappings to preserve optimal solutions of projection problems in semiring-induced valuation algebra systems. We show that, for a mapping between two semiring-induced valuation algebras, its properties of preserving optimal solutions and preserving the solution ordering are equivalent, if it is order-reflecting. Further, we propose some necessary and sufficient conditions for a projection problem and its translation to have a same optimal solution set. Finally, we show that a lower adjoint of a Galois connection defined between c-semirings preserves optimal solutions of projection problems, if it is order-preserving and surjective.