Conical decomposition and vector lattices with respect to several preorders (Q2495224)

Consider \(n\) \((n \geq 2)\) convex cones \(K_1, \dots, K_n\) in a vector space \(E\). Let \(L = \sum_{i=1}^n K_i\) be the sum of these cones. A collection of elements \(x_i \in K_i \) \((i = 1, \dots, n)\) is called the decomposition of an element \(x \in L\) with respect to \(\{K_i\}\) if \(x = x_1 + x_2 + \dots + x_n\). The paper is concerned with the totality of all possible decompositions for all \(x \in L\). The authors consider the set-valued mapping \(\sigma\) defined on \(L\) by \[ \sigma (x) = \{(x_1, \dots, x_n), \quad x_i \in K_i, \quad \sum_{i=1}^n x_i = x\}. \] The mapping \(\sigma\) is called the decomposition mapping with respect to the cones \(K_1, \dots, K_n\). They study conditions that provide the additivity of the decomposition mapping. For this purpose, they introduce and study the Riesz interpolation property and lattice properties of spaces with respect to several preorders. The notion of 2-vector lattice is introduced and investigated. Theorems that give the relationship between the Riesz interpolation property and lattice properties of the dual spaces are also given.