On nonlinear functional spaces based on triangular conorms

Abstract

a b s t r a c tIn this paper, we discuss the nonlinear functional spaces based on triangular conorms. Particularly, we discuss the properties of the upper-closures of the regular subspaces of the nonlinear functional space based on a continuous triangular conorm. Furthermore, we prove that with respect to a strict triangular conorm, a subset of the nonlinear functional space is an upper-complete normal subspace if and only if the family of all sets whose characteristic functionals are contained in the given subset of the nonlinear functional space is a sigma-algebra.

On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

“…We will denote by Δ maximum element on [ , ] (usually Δ is either or ) with respect to this total order. Definition 1 (see [34]). Let { } be a sequence from [ , ].…”

“…Definition 7 (see [34]). For any continuous pseudoaddition ⊕, if , ℎ ∈ F( ), then define the paracomplement set | − ⊕ ℎ| as the set of all those functionals such that…”

“…Definition 8 (see [34]). For any strict pseudoaddition ⊕ and , ∈ [ , ] with ⪯ , the complement − ⊕ is defined as…”

“…Definition 10 (see [34]). For any pseudoaddition ⊕, a nonempty subset K of F( ) is said to be a functional space with respect to ⊕, denoted by (K, ⊕), if ( ⊙ ) ⊕ ( ⊙ ℎ) ∈ K for all , ℎ ∈ K and , ∈ [ , ], where ⊙ is a distributive pseudomultiplication with respect to ⊕.…”

“…If (K, ⊕) is a functional space with respect to ⊕, then we just write K instead of (K, ⊕) whenever ⊕ can be determined from the context. Definition 11 (see [34]). For each subset A of F( ) the upper closure of A, denoted byÂ, is the set of all elements of F( ) having the form lim → ∞ for some increasing sequence { } ≥1 from A.…”