Abstract
In this paper, we essentially deal with Köthe-Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh's extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions. In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Finally, we obtain the Köthe-Toeplitz duals of fuzzy level sets with respect to the partial metric based on a partial ordering.
Highlights
By ω(F), we denote the set of all sequences of fuzzy numbers
We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets
Partial metrics are more flexible than metrics; they generate partial orders and their topological properties are more general than the one for metrics, argued by the fact that the self-distance of each point need not be zero
Summary
By ω(F), we denote the set of all sequences of fuzzy numbers. We define the classical sets bs(H), cs(H), and cs0(H) consisting of the sets of all bounded, convergent, and null series, respectively; that is n bs (H) := {u = (uk) ∈ ω (F) : (∑uk) ∈ l∞ (H)} , k=0 n cs (H) := {u = (uk) ∈ ω (F) : (∑uk) ∈ c (H)} , (1). Where the distance function Hs denotes the partial metric of fuzzy level sets defined by. (c) A partial metric space (X, p) is said to be complete if every Cauchy sequence (xn) in X converges, with respect to the topology τp, to a point x ∈ X such that p(x, x) = limm,n → ∞ p(xm, xn). (e) A sequence (xn) in a partial metric space (X, p) converges to a point x ∈ X, for any ε > 0 such that x ∈ Bp(x, ε), there exists n0 ≥ 1 so that for any n ≥ n0, xn ∈ Bp(x, ε). The fuzzy number uk denotes the value of the function at k ∈ N and is called as the general term of the sequence. ∈ ω(F) is bounded the sequences and {uk+(λ)} are uniformly bounded in [0, 1]
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