SOME RESULTS RELATED WITH FUZZY a −NORMED LINEAR SPACE

Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, 𝑀𝑛(𝐹) denotes the set of all fuzzy matrices of order 𝑛 over the fuzzy unit interval [0,1]. Inaddi tion (𝑀𝑛 (𝐹), 𝜃) dis called as fuzzy 𝛼 −normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy 𝛼 −normed linear space. The set of all fuzzy points in 𝑀𝑛 (𝐹) is denoted by 𝑃∗(𝑀𝑛(𝐹)). For a fuzzy 𝛼 −normed linear space (𝑀𝑛 (𝐹), 𝜃), we have |𝜃(𝑃𝐴)𝛼 −𝜃(𝑃𝐵)𝛼 | ≤ 𝜃(𝑃𝐴,𝑃𝐵)𝛼. Besides 𝜃 is a continuous function on 𝑀𝑛 (𝐹). That is, if 𝑃𝐴𝑛 → 𝑃𝐴 as 𝑛 → ∞ then 𝜃(𝑃𝐴𝑛 )𝛼 → 𝜃(𝑃𝐴)𝛼 as 𝑛 → ∞, where 𝑃𝐴𝑛 is a sequence in (𝑀𝑛 (𝐹), 𝜃). Hence, 𝜃 is always bounded on 𝑀𝑛(𝐹). Next we introduce the following result: Let 𝑃𝐴𝑛 , 𝑃𝐵𝑛 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)) with 𝑃𝐴𝑛 and 𝑃𝐵𝑛 converge to 𝑃𝐴 and 𝑃𝐵 respectively as 𝑛 → ∞. Then 𝑃𝐴𝑛 + 𝑃𝐵𝑛 converge to 𝑃𝐴 + 𝑃𝐵 as 𝑛 → ∞. Furthermore, we are able to compare two different fuzzy 𝛼 −norms with convergent sequence. The result states that for a fuzzy 𝛼 −normed linear space (𝑀𝑛(𝐹), 𝜃), we have 𝜃(𝑃𝐴)𝛼1 ≥ 𝑀𝜃(𝑃𝐴 )𝛼2 , for some 𝑀 > 0 and 𝑃𝐴 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)). If 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼1 −norm then 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼2 −norm. Moreover, if (𝑀𝑛 (𝐹), 𝜃) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy 𝛼 −normed linear space and its properties.

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Single-Point Characterization of Uninorms with Nilpotent Underlying T-Norm and T-Conorm

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488514500299 ◽

2014 ◽

Vol 22 (04) ◽

pp. 591-604 ◽

Cited By ~ 32

Author(s):

Gang Li ◽

Hua-Wen Liu ◽

János Fodor

Keyword(s):

Set Theory ◽

Fuzzy Set ◽

Fuzzy Set Theory ◽

Single Point ◽

Unit Interval ◽

Binary Operations ◽

Construction Methods

Uninorms, as binary operations on the unit interval, have been widely applied in fuzzy set theory. In this paper, we study uninorms with nilpotent underlying t-norm and t-conorm. We prove that such a uninorm belongs to [Formula: see text] or [Formula: see text]. Moreover, some construction methods of uninorms from given t-norm and t-conorm are discussed.

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Distributivity for 2-Uninorms Over Semi-Uninorms

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488517500131 ◽

2017 ◽

Vol 25 (02) ◽

pp. 317-345 ◽

Cited By ~ 4

Author(s):

Ya-Ming Wang ◽

Feng Qin

Keyword(s):

Set Theory ◽

Fuzzy Set ◽

Fuzzy Set Theory ◽

Unit Interval ◽

The Other ◽

Fuzzy Implications ◽

Logical Connectives

This paper is mainly devoted to solving the distributivity equations for 2-uninorms over semi-uninorms. Our investigations are motivated by the couple of distributive logical connectives and their generalizations, such as t-norms, t-conorms, uninorms, nullnorms, and fuzzy implications, which are often used in fuzzy set theory. Recently, there are their two generalizations. One is a 2-uninorm covering both a uninorm and a nullnorm, which forms a class of commutative, associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. The other is a semi-uninorm, which generalizes a uninorm by omitting commutativity and associativity. In this work, all possible solutions of the distributivity equation for the three defined subclasses of 2-uninorms over semi-uninorms are characterized.

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