Outer and inner approximations in quantum spaces

Abstract

Abstract The present paper introduces and studies the concepts of K-outer approximation and K-inner approximation for a monotone function μ defined on a D-poset P, by a subfamily K of P. Some desirable properties of K-approximable functions are established and it is shown that the family of all elements of P that possess K-approximation, forms a lattice and is closed under orthosupplementation. We have proved that a submodular measure on a suitable subfamily of P having K-outer approximation can be extended to a function that has K-outer approximation, and a tight function that has K-inner approximation can be extended to a function having K-inner approximation.