Extensions of Partial Lattice-Valued Possibilistic Measures From Nested Domains

Abstract

We investigate a partial non-numerical possibilistic measure taking its values in a complete lattice and defined on a nested system of subsets of the universe under consideration. Our aim is to extend this measure conservatively to the power-set of all systems of this universe using the same idea as that when introducing outer measures. Hence, we ascribe to each subset of the universe its minimal (in the sense of set inclusion) covering by a set from the nested domain and define its possibility degree as identical with the value ascribed to this covering. Also analyzed is the case when they are two lattice-valued possibilistic measures on the same universe, each with its own nested domain and our aim is to define one possibilistic measure on the power-set in question in a way sophistically taking profit of the information offered by both the particular partial lattice-valued measures.

Several results on set-valued possibilistic distributions

“…In [7], we proposed possibilistic distributions π : Ω → P(X) and possibilistic measures Π : P(Ω) → P(X) as complete mappings, so that for each ω ∈ Ω and each A ⊂ Ω the values π(ω) ∈ P(X) and Π(A) = ω∈A π(ω) are defined. However, only the value P ( A∈S A) for a finite system S of a disjoint subsystem of A may be defined and computed from values of P on A in probability spaces Ω, A, P with finitely additive probability measure P on a finite field A.…”

“…Inspired by Lemma 2.4 and Lemma 4.1, we propose in [7,8,9] some modifications of the space of values in which the mapping π : Ω → T takes its values in such a way that π(ω 0 ) = 1 T is valid for only one ω 0 ∈ Ω. In [7], the mapping π, defined on Ω, takes its values in a complete chained lattice; let us recall, for the reader's convenience, the way leading to this notion.…”

“…In [7], the mapping π, defined on Ω, takes its values in a complete chained lattice; let us recall, for the reader's convenience, the way leading to this notion.…”

“…The values of possibilistic distributions are taken from complete lattices in [7], but they are bound by the condition of chained structure, so that each two possibility degrees are comparable by the partial ordering relation ≤ defined on T = T, ≤ . In what follows, we use a more intuitive space of values, namely, that of the power-set over the space X.…”

“…In what follows, we use a more intuitive space of values, namely, that of the power-set over the space X. However, the conditions imposed on chained lattices need not be valid in general; that is, the structures from [7] and that introduced in this text cannot be classified as being a particular case or, in contrary, a generalization of each other. Theorem 4.2.…”