Type-2 implications on non-interactive fuzzy truth values

2017

Fuzzy Sets and Systems

Cubillo et al. in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset (partially ordered set), in particular on M (set of fuzzy membership degrees of T2FSs, which are the functions from [0, 1] to [0, 1]). Previously, Taká cˇ in 2014 had applied Zadeh's extension principle to obtain a set of operators on M which are, under some conditions, aggregation operators on L*, the set of strongly normal and convex functions of M. In this paper, we introduce a more general set of operators on M than were given by Taká cˇ, and we study, among other properties, the conditions required to satisfy the axioms of the aggregation operator on L (set of normal and convex functions on M), which includes the set L*.

“…Definition 6. ( [13,30,10,11]) The operations of u (generalized maximum), n (generalized minimum), ¬ (complementation) and the elements 0 and 1 are defined on M as follows:…”

Section: Several Types Of Fuzzy Sets and Operationsmentioning

confidence: 99%

Aggregation operators on type-2 fuzzy sets

2017

Fuzzy Sets and Systems

“…Definition 7. [36,38,39] In M, the operations (generalized maximum), (generalized minimum), ¬ (complementation) and the elements0 and1 are defined as follows:…”

Section: Examplementioning

confidence: 99%

New Order on Type 2 Fuzzy Numbers

et al. 2017

Axioms

Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers.

“…Definition 6. ( [7], [19], [4], [5]) The operations of ⊔ (generalized maximum), ⊓ (generalized minimum), ¬ and the elements0 y1 are defined on M as follows:…”

Section: Preliminariesmentioning

confidence: 99%

Examples of Aggregation Operators on Membership Degrees of Type-2 Fuzzy Sets

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

2015

Z. Takáč in [16] introduced the aggregation operators on any subalgebra of M (set of all fuzzy membership degrees of the type-2 fuzzy sets, that is, the functions from [0,1] to [0,1]). Furthermore, he applied the Zadeh's extension principle (see [24]) to obtain in [16,17] a set of aggregation operators on L* (the strongly normal and convex functions of M). In this paper, we introduce the aggregation operators on any partially ordered and bounded set (poset). This will allow us to suitably provide aggregation operators on M. In this sense, firstly we define a set of operators on M, more general than those given by Z. Takáč, studying some of their properties. Secondly, we focus on some operators obtained through a very different way, proving that they are aggregation operators on L (set of normal and convex functions of M), and on M.