Search

In numerous generalizations of the original theory of fuzzy sets proposed by Zadeh, the considered membership degrees are often taken from lattices isomorphic to the lattice L I of closed subintervals of the unit interval [ 0 , 1 ]. This is, for example, the case of intuitionistic values, Pythagorean values or q -rung orthopair values. The mentioned isomorphisms allow to transfer the results obtained for the lattice L I directly to the other mentioned lattices. In particular, basic connectives in Zadeh's fuzzy set theory, i.e., special functions on the lattice [ 0 , 1 ] , can be extended to the interval-valued connectives, i.e., to special functions on the lattice L I , and then to the connectives on the lattices L ⁎ of intuitionistic values, P of Pythagorean values, and also on the lattice L τ q of q -rung orthopair values. We give several examples of such connectives, in particular, of those which are related to strict t-norms. For all these connectives we show their link to an additive generator f of the considered strict t-norm T. Based on our approach, many results discussed in numerous papers can be treated in a unique framework, and the same is valid for possible newly proposed types of connectives based on strict t-norms. Due to this approach, a lot of tedious proofs of the properties of the proposed connectives could be avoided, which gives researchers more space for presented applications. [ABSTRACT FROM AUTHOR]

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this article for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt, ordinal sums of triangular norms on bounded lattices, Inf. Sci., vol. 517, (2020) 198–216. [ABSTRACT FROM AUTHOR]

Some relevant notions in fuzzy set theory are those of triangular-(t)- norm, t-conorm, and negation, which provide a systematic way of defining set-theoretic operations, or, from other point of view, logical connectives. For instance, the majority of fuzzy implications are directly derived from these operators, so they play a prominent role in fuzzy control theory or in approximate reasoning. This incites the search of suitable t-norms, t-conorms, and negations for solving each specific problem. In this article, we propose a procedure, that we call induction, for designing them on spaces of lattice-valued maps. Concretely, for each family of operators (t-norms, t-conorms, or negations) indexed in the domain set, we may induce an operator of the same kind, so that our method offers a great flexibility in the design task. It may be applied to well-known fuzzy objects as interval-valued or type-2 fuzzy sets. Nevertheless, the theory is formally developed for arbitrary bounded lattices. [ABSTRACT FROM AUTHOR]

Recently the distributive equations involving various classes of aggregation operators have aroused widespread attention because of their importance in the theoretic and applied communities of fuzzy set theory. 2-uninorms and semi-t- operators are two special classes of aggregation operators and have been proved to be useful in many areas such as fuzzy decision making, approximate reasoning and so on. Therefore, the aim of this paper is to investigate the left (right) distributivity of semi-t-operators over 2-uninorms. We consider five subclasses of 2-uninorms and characterize the corresponding left (right) distributive equations of semi-t-operators over 2-uninorms. [ABSTRACT FROM AUTHOR]

It is well known that under some conditions, a convex combination of the drastic product T D and the strict triangular norm is still a triangular norm. Uninorms, as a special kind of generalization of both triangular norms and triangular conorms, have been widely used in fuzzy set theory. In this paper, the convex combination of uninorms is studied. Some properties of the class of uninorms with the underlying triangular norm T D are presented firstly. Then the convex combination of this class of uninorms is discussed and it is shown that only in some trivial cases, the convex combination is still a uninorm. [ABSTRACT FROM AUTHOR]

Measuring the similarity between images is an essential problem in various image processing and pattern recognition applications. In pattern recognition problems, it is indispensable to give formulas for calculating similarity between different patterns. But it is very difficult to find a certain measure that can be successfully applied to all kinds of pattern recognition problems. Intuitionistic fuzzy sets have been successfully applied to various areas such as pattern recognition and medical diagnostics. In intuitionistic fuzzy sets theory, the calculation of the similarity between intuitionistic fuzzy sets is a significant technique for distinguishing the similarity degree between intuitionistic fuzzy sets. The existing similarity measures almost are obtained in the sense of distance. In this paper, we present a novel way to obtain the similarity measure between intuitionistic fuzzy sets from a new perspective. Our main purpose is to show that according to the membership and non-membership functions of intuitionistic fuzzy sets, a triangular norm can induce an inclusion degree. Using this triangular norm and the induced inclusion degree, a similarity measure of intuitionistic fuzzy sets can be obtained. We also prove some properties of the proposed similarity measure between intuitionistic fuzzy sets. As the applications of similarity degree proposed in this paper, we first present an intuitionistic fuzzy clustering algorithm based on similarity degree. Then, the similarity degree proposed in this paper is applied to pattern recognition. At the same time, the numerical examples are employed to illustrate the effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

In this study, based on the knowledge of the existence of t-norms on an arbitrary given bounded lattice, we introduce t-closure operators with the help of a t-norm on the lattice and a subset of the lattice including the top element. We define two equivalence relations by using t-closure operators. The first one is on the set of all t-norms on a bounded lattice. An important class is obtained according to this relation. We define a partially order on the set of all equivalent relations given as secondly. Further, we define a set, denoted by , and by using this set, we investigate under which conditions L can be embedded into . We obtain a topology by using t-closure operators and examine some properties of this topology. Lastly, we generalize partial order. [ABSTRACT FROM AUTHOR]

Recently, a class of uninorms locally internal on the boundary was proposed by Mas et al. during the study of the migrativity property for uninorms. In this paper, some properties of this class of uninorms are presented and the characterizations are obtained for different cases. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to solving the functional equation of distributivity between aggregation operators with 2-neutral element. Our investigations are motivated by the couple of distributive logical connectives and their generalizations used in fuzzy set theory e.g., triangular norms, conorms, uninorms, nullnorms and implications. One of the recent generalizations covering both uninorms and nullnorms are 2-uninorms, which form a class of commutative, associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. In this work the distributivity of two binary operators from the class of 2-uninorms is considered. In particular, all possible solutions of the distributivity equation for the three defined subclasses of these operators depending on the position of its zero and neutral elements are characterized. [ABSTRACT FROM AUTHOR]

In the present paper, definitions of an n -ary semicopula on a bounded lattice L , a ∨-distributive n -ary semicopula on L and an infinite ∨-distributive n -ary semicopula on L are introduced. A relation between the set of all bounded lattices L with a ∨-distributive n -ary semicopula on L and the set of all bounded lattices L with a ∨-distributive m -ary semicopula on L , where 2 ≤ n , 2 ≤ m , n ≠ m , is obtained. The method is given for investigation of the following questions: a description of all bounded lattices L such that every n -ary semicopula on L is ∨-distributive and a description of all complete lattices L such that every n -ary semicopula on L is infinite ∨-distributive. [ABSTRACT FROM AUTHOR]

In this paper, an equivalence relation on the class of t-norms induced by a T -partial order is provided and discussed. The equivalence classes linked to some special t-norms are characterized as well as some properties preserved by the introduced equivalence. Defining the set of incomparable elements with respect to the T -partial order, this set is deeply investigated. Finally, we give an answer to a recently posed open problem. [ABSTRACT FROM AUTHOR]

Abstract: T-product extensions (where T is a continuous t-norm) of a system of low-dimensional possibility distributions form an important class of solutions of the so-called possibilistic marginal problem. Nevertheless, they can differ from each other, e.g., from the viewpoint of inference. Therefore the need for their interpretation is obvious. To find it, we identify any possibility distribution with a set of probability distributions dominated by it and find a probability interpretation of models based on Gödel's, product and Łukasiewicz's t-norms. [Copyright &y& Elsevier]

Abstract: One of the most important operators in soft computing are the triangular norms (t-norms), as well as the combination among them. The ordinal sum of triangular norms on [0, 1] has been used to construct other triangular norms. However, on a bounded lattice, an ordinal sum of t-norms may not be a t-norm. Several necessary and sufficient conditions are presented in this paper for ensuring whether an ordinal sum on a bounded lattice of arbitrary t-norms is, in fact, a t-norm. Moreover, we show that a large set of ordinal sums verify these conditions. Hence, they are very interesting in order to verify whether an ordinal sum on a bounded lattice is a t-norm for a particular family of t-norms. [Copyright &y& Elsevier]

Abstract: In this paper we continue the study of a submeasure notion introduced in Hutník and Mesiar (2009) involving a class of operations which provides a generalization of-submeasures. We construct pseudo-metrics and metrics generated by such probabilistic submeasures. Two possible generalizations of our submeasure notion are discussed. [ABSTRACT FROM AUTHOR]

Using the theory of countable extension of t-norm we prove a common fixed point theorem for compatible mappings satisfying an implicit relation in fuzzy metric spaces. [ABSTRACT FROM AUTHOR]

It is presented an approach to decision theory based upon nonprobabilistic uncertainty. There is an axiomatization of the hybrid probabilisticpossibilistic mixtures based on a pair of triangular conorm and triangular norm satisfying restricted distributivity law, and the corresponding non-additive Smeasure. This is characterized by the families of operations involved in generalized mixtures, based upon a previous result on the characterization of the pair of continuous t-norm and t-conorm such that the former is restrictedly distributive over the latter. The obtained family of mixtures combines probabilistic and idempotent (possibilistic) mixtures via a threshold. [ABSTRACT FROM AUTHOR]

Abstract: The lattice-theoretic structure of the set of semi-copulas and some of its subsets is investigated. As a relevant case, we obtain that the set of commutative semi-copulas is a lattice completion of the set of triangular norms. [Copyright &y& Elsevier]

We consider a property on binary operations, which we call -migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous. [Copyright &y& Elsevier]

Abstract: Recently, M. Urbański and J. Wa̧sowski [Fuzzy arithmetic based on boundary weak t-norms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 13 (2005) 27–37] extended the triangular norm operators to a new class of operators called the boundary weak triangular norms. This extension is based on the replacement of the condition by the weaker one for . In this paper we discuss the construction of the boundary weak triangular norms based on their additive generators. Continuous decreasing functions which generate proper boundary weak triangular norms are completely characterized. Several examples are included. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new method for constructing non-continuous t-norms from a given t-norm. Using this method, we construct a family of conditionally cancellative left-continuous t-norms which are not continuous. Thus we answer an open question which is collected by Klement et al. [Problems on triangular norms and related operators, Fuzzy Sets and Systems 145 (2004) 471–479]. [Copyright &y& Elsevier]

Abstract: Using the infinitely countable extension of triangular norms, a generalization of Tardiff''s fixed point theorem in probabilistic metric spaces is proved. As a consequence, an application to random equations is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, several methods to construct triangular norms on partially ordered sets are proposed. These methods include: [(1)] Constructing triangular norms on the poset of closed intervals of a given bounded poset. [(2)] Constructing triangular norms via Galois connections between bounded posets based on a categorical analysis of the construction of triangular norms on closed intervals via monotone functions and their pseudo-inverses. [(3)] Constructing triangular norms on function spaces between partially ordered sets. [Copyright &y& Elsevier]

Abstract: An open problem on triangular norms, recently posed by Klement et al. [Fuzzy Sets and Systems 145 (2004) 471], is discussed in the class of associative copulas. [Copyright &y& Elsevier]

Compactness is an important property of classical logic. It states that simultaneous satisfiability of an infinite set of formulas is equivalent to the satisfiability of all its finite subsets. In fuzzy logics, we have different degrees of satisfiability, hence the questions of compactness become more complicated. Here we give an overview of the recent results on compactness and we extend them to various fuzzy logics. [Copyright &y& Elsevier]

We give an improvement of Theorem 1 from [2] with a quite different approach, which enable us to prove that the fixed point is also globally attractive. In Theorem 2.11 a further generalization is obtained for a complete Menger space (S,F,T), where T belongs to a more general class of continuous t-norms than in the previous case where T=TM (=min). Theorem 3.2 is a generalization of Theorem 2 from [2]. Thereafter the notion of a generalized C-contraction of Krasnoselski's type is introduced and a fixed point theorem for such mappings is proved. An application in the space S(Ω, K, P) is given. [ABSTRACT FROM AUTHOR]

In order to express the degree to which a subset of a finite universe is contained into another subset, the concept of inclusion measure (or subsethood measure) of ordinary sets is introduced. A distinction is made between three types of inclusion measures. The first type yields reflexive inclusion measures, whereas the second and third type both give rise to locally reflexive inclusion measures, the latter ones simply being complementary to the former ones.Furthermore, a systematic way of generating inclusion measures for ordinary sets is presented in the form of a rational expression solely based on cardinalities of the sets involved. Various properties of the obtained rational inclusion measures, such as monotonicity and transitivity, are investigated. [Copyright &y& Elsevier]

The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh's quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve ⊗ D − convexity and ⊗ a − convexity on bounded lattices, ⊗ M preserves ⊗ H − convexity in the real unite interval [ 0 , 1 ] . [ABSTRACT FROM AUTHOR]