Search

We introduce the concept of intuitionistic fuzzy BCK-submodules of a BCK-module with respect to a t-norm and a s-norm and present some basic properties. [ABSTRACT FROM AUTHOR]

The idempotent uninorms are characterized by means of the ordinal sum of Clifford. It is shown that idempotent uninorms are in one-to-one correspondence with special linear orders on [ 0 , 1 ] . A connection between respective linear order on [ 0 , 1 ] and the characterizing multi-function of the uninorm is also investigated. [ABSTRACT FROM AUTHOR]

Triangular norms and conorms with continuous diagonals are discussed. In literature we can find examples of non-continuous t-norms with continuous diagonals, however, their deeper study is still missing. In this paper we introduce a sufficient condition under which a t-norm with continuous diagonal is continuous. Moreover, we show that an Archimedean t-norm continuous on the boundary is continuous. Several illustrative examples are also included. [ABSTRACT FROM AUTHOR]

The aim of this paper is to complete some results in the paper “Migrative uninorms and nullnorms over t-norms and t-conorms” [1] . That paper studied the alpha-migrativity of uninorms over t-norms. Here, migrativity in the other direction is investigated. [ABSTRACT FROM AUTHOR]

In the framework of complete probabilistic Menger metric spaces, this paper investigates some relevant properties of convergence of sequences built through sequences of operators which are either uniformly convergent to a strict k-contractive operator, for some real constant k ∈ (0, 1), or which are strictly k-contractive and point-wisely convergent to a limit operator. Those properties are also reformulated for the case when either the sequence of operators or its limit are strict $$\varphi$$ -contractions. The definitions of strict ( k and $$\varphi$$ ) contractions are given in the context of probabilistic metric spaces, namely in particular, for the considered probability density function. A numerical illustrative example is discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study possible multivariate generalizations of construction of n-copulas with a given diagonal section. In the case of Archimedean n-copulas we prove that Frankʼs condition, which establishes when an Archimedean 2-copula is characterized by its diagonal section, can be extended to larger dimensions. [Copyright &y& Elsevier]

Abstract: In order to avoid combinatorial rule explosion in fuzzy reasoning, Qin and Baczyński, in [16], investigated the distributivity equation of implication , when is a continuous but not Archimedean triangular norm, is a continuous and Archimedean triangular norm and I is an unknown function. In fact, it partially answered the open problem suggested by Baczyński and Jayaram in [5]. In this work we continue to explore the distributivity equation of implication , when both and are continuous but not Archimedean triangular conorms, and I is an unknown function. Here it should be pointed out that these results make difference with recent ones obtained in [16]. Moreover, our method can still apply to the three other functional equations related closely to this equation. It is in this sense that we have completely solved the open problem commented above. [Copyright &y& Elsevier]

The evaluation and expression of uncertainty in measurement is one of the fundamental issues in measurement science and challenges measurement experts especially when the combined uncertainty has to be evaluated. Recently, a new approach, within the framework of possibility theory, has been proposed to generalize the currently followed probabilistic approach. When possibility distributions are employed to represent random contribution to measurement uncertainty, their combination is still an open problem. This combination is directly related to the construction of the joint possibility distribution, generally performed by means of t-norms. The problem is that joint possibility distributions strongly depend on the considered t-norm, and, therefore, the choice of one particular t-norm over another has to be justified. The first goal of this paper is, hence, to provide support to the choice of a particular t-norm. The second goal is to discuss the construction of the joint possibility distribution when the two random contributions to uncertainty show mutual dependence, with specific reference to a particular class of possibility distributions. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we define (α, β)-equalities of intuitionistic fuzzy sets and (θ α , ϕ β )-equalities of T, S-residual implications and their algebraic operations. Meanwhile (α, β)-equalities with respect to intuitionistic fuzzy relations, extension principle and dual triangular norms are studied. In addition, we discuss the intuitionistic fuzzy choice functions and their properties. Finally, a ordered method of intuitionistic fuzzy choice functions is proposed by using some examples. [Copyright &y& Elsevier]

We consider the generalized fuzzy integral introduced by Fang[4) and use the concept of interval-valued functions which is used for representing uncertain functions. In this paper, we define the interval-valued generalized fuzzy integral with respect to a fuzzy measure by means of an inter val-representable generalized triangular norm of measurable interval-valued functions and investigate some characterizations and convergence properties of them. [ABSTRACT FROM AUTHOR]

Using Generalized Modus Ponens reasoning, we examine the values of the inferred conclusion depending on the correspondence between the premise of the rule and the observed fact. The conclusion is obtained using a set of implications in order to represent a fuzzy if-then rule with a single input single output and the t-norm with threshold generated by t-norm T( x, y) = xy, as a compositional operator. Some comments and an example are presented in order to show how the obtained results can be used. [ABSTRACT FROM AUTHOR]

Abstract: First, we show that the class of left-continuous triangular norms whose associated negations are discontinuous only at one point is not generally continuous except for the trivial case. Second, we find the strongest left-continuous triangular norm with a given associated negation and shows that there is no weakest left-continuous triangular norm with a given associated negation. [Copyright &y& Elsevier]

Abstract: The -migrative triangular subnorms, where T stands for the three prototype triangular norms (namely the product, the Łukasiewicz t-norm and the minimum), are investigated in detail. The paper gives necessary and sufficient conditions under which a triangular subnorm with a continuous additive generator is -migrative with respect to any of the three prototype triangular norms. [Copyright &y& Elsevier]

Abstract: A new class of fuzzy implications, called -implications, is introduced by means of generalized h-generators. Basic properties of these implications are discussed. It is shown that the -implications are really a new class different from the known (S,N)-, R-, QL- and Yager''s f- and g-implications. Generalizations of three classical logic tautologies with implications, viz., law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are established, under which the corresponding functional equations are satisfied. [Copyright &y& Elsevier]

Abstract: This paper generalizes the migrativity of triangular norms (t-norms for short). Precisely, we find those continuous t-norms T which satisfy , where are given. It is worth pointing out that this paper provides a new characterization of the product t-norm. [Copyright &y& Elsevier]

An isomorphism f between two continuous Archimedean t-norms T and T′ transforms a T-indistinguishability operator E into a T′-indistinguishability operator f ∘ E and many interesting properties of E are transfered to f ∘ E by f. This paper generalizes this result in order to relate indistinguishability operators with respect to two non isomorphic continuous Archimedean t-norms. This will allow us to transfer definitions and properties from strict to non-strict Archimedean t-norms and vice versa. [ABSTRACT FROM AUTHOR]

Abstract: Standby redundant system is a powerful tool to enhance the reliability, availability, quality, and safety of operational plants. This study proposed an approximate Tω (the weakest t-norm) fuzzy Graphical Evaluation and Review Technique (GERT) simulation technology to simulate two-unit standby redundant system reliability. The approximate fuzzy arithmetic operations employ principle of interval arithmetic under the Tω arithmetic operations. Therefore, the novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating, under vague environment. In numerical examples the approximate fuzzy arithmetic operations has evidenced that it can successfully calculate results of fuzzy operations as interval arithmetic, and can more effectively reduce fuzzy spreads. In the real fuzzy repairable reliability model the performance also shows that the approximate fuzzy arithmetic operations successfully simulate/analyze the two-unit standby redundant system reliability and obtain more confident fuzzy results under vague environment. [Copyright &y& Elsevier]

Abstract: In this work smooth t-subnorms and boundary weak t-norms (bwt-norms) defined on a finite chain are studied. In the smooth case, it is proved that smooth bwt-norms reduces to smooth t-norms, whereas a classification of smooth t-subnorms as ordinal sums of a smooth t-norm and an Archimedean smooth t-subnorm is proved, in a similar way as in the case of [0,1]. With respect to Archimedean smooth t-subnorms, some cases depending on their zero-region are considered, classifying all of them in some cases and some specific families in others. [Copyright &y& Elsevier]

Abstract: Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system. [ABSTRACT FROM AUTHOR]

Alsina, Frank, and Schweizer posed a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. In this paper we deal with the class of strict t-norms with well-defined first partial derivatives along their zero borders. We show that the answer to the question is negative for certain couples of such t-norms and we state out possible further research. [ABSTRACT FROM AUTHOR]

The main results of the present paper are related to the use of finite-difference operators for estimating the norms of inverses of differential operators with unbounded operator coefficients. We obtain a new proof of the Gearhart-Prüss spectral mapping theorem for operator semigroups in a Hilbert space and estimate the exponential dichotomy exponents of an operator semigroup. [ABSTRACT FROM AUTHOR]

Using Generalized Modus Ponens reasoning, we examine the values of the inferred conclusion depending on the correspondence between the premise of the rule and the observed fact. The conclusion is obtained using Fodor's implication in order to represent a fuzzy if-then rule with a single input single output and the t-norm with threshold generated by t-norm product, as a compositional operator. A comparison study with the case when the standard t-norm product is used is made. Some comments and an example are presented in order to show how the obtained results can be used. [ABSTRACT FROM AUTHOR]

This paper examines the practical construction of κ-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on κ-convex additive generators and translate κ-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees κ-Lipschitz properly of the resulting triangular norms and conorms. [ABSTRACT FROM AUTHOR]

In this paper we study a generalization of a submeasure notion which is related to a probabilistic concept, especially to Menger spaces where triangular norms play a crucial role. The resulting notion of a τT-submeasure is suitable for modeling those situations in which we have only probabilistic information about the measure of the set. We characterize a class of universal τT-submeasures (i.e., τT-submeasures for an arbitrary t-norm T) and give explicit formulas for τT-submeasures for some classes of t-norms. Also, transformations and aggregations of τT-submeasures are discussed. [ABSTRACT FROM AUTHOR]

This work concerns nonstrict Archimedean triangular norms and cardinalities of fuzzy sets. Values of those t-norms are expressed in the language of Hamming distances between some fuzzy sets. We apply this optics to two important types of cardinalities, namely generalized FGCounts and generalized sigma counts. Finally, we prove that the well-known relationship between the FGCount and the sigma count of a fuzzy set is a limit property of generalized FGCounts involving Yager t-norms. © 2009 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, ..., +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, ..., +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic. [ABSTRACT FROM AUTHOR]

Abstract: Focusing on conjunctive, left-continuous, increasing functions T we redefine the rotation invariance property in terms of contour lines. Under the assumption of the existence of a neutral element , this rotation invariance property requires some partial commutativity and associativity. The functions that are rotation invariant w.r.t. all of their contour lines are characterized. [Copyright &y& Elsevier]

Abstract: Uninorms are an important generalization of triangular norms and conorms, having a neutral element lying anywhere in the unit interval. In this paper, we introduce the concepts of left and right uninorms on a complete lattice. We discuss the residual operations of left and right uninorms, and study some basic properties of the residual operations of infinitely -distributive left (right) uninorms and pseudo-uninorms. [Copyright &y& Elsevier]

Abstract: Fono and Gwét [On strict lower and upper sections of fuzzy orderings, Fuzzy Sets and Systems 139 (2003) 583–599] introduced and studied strict lower section, strict upper section and strict order interval of a given weakly complete fuzzy pre-order R when the minimal regular fuzzy strict component of R is defined by the residual coimplicator of the ukasiewicz t-conorm. And they gave an example of application of fuzzy strict lower sections in economics. In this paper, we generalize their framework for the minimal regular fuzzy strict component of R defined by the residual complicator of a continuous t-conorm. We show that fuzzy strict sections and fuzzy strict order intervals, defined in this general case, generate fuzzy topologies. [Copyright &y& Elsevier]

Abstract: Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference generator, in particular a Frank t-norm. In the weakly complete case, we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them. In the strongly complete case, we lay bare a stronger type of transitivity of the strict preference relation. We conclude the paper with a rather negative result: there is no hope to obtain a compositional characterization of weakly complete fuzzy pre-orders, and hence also not of fuzzy pre-orders in general. [Copyright &y& Elsevier]

Abstract: Convergence of sequences of triangular norms and their quasi-inverses, stability of the representation theorem for T-similarity relations, and the density property of the family of Z-similarity relations are investigated. It is shown, in particular, that under some hypothesis the representation theorem is stable under taking limits of continuous t-norms, and that any strictly reflexive and symmetric fuzzy relation can be approximated up to any precision level by a T-similarity relation with T being a strict t-norm. [Copyright &y& Elsevier]

The notion of T-fuzzy Hν-submodules of Hν-modules is introduced and basic properties are investigated by means of the use of a triangular norm T. Using a collection of Hν-submodules of Hν-modules, we establish T-fuzzy Hν-submodules of Hν-modules. Moreover, the notion of Notherian Hν-modules is introduced, and its characterization is given. At last, the properties of T-fuzzy characteristic Hν--submodules are discussed. [ABSTRACT FROM AUTHOR]

Abstract: Recently the distributivity equation was investigated in families of certain operations (e.g. triangular norms, conorms, uninorms and nullnorms). Generally, the assumption of associativity is not necessary in consideration of distributivity equation. Moreover, if we omit commutativity assumption, consideration of the left and right distributivity conditions is reasonable. Previous results about distributivity between uninorms and nullnorms can be obtained as simple corollaries. [Copyright &y& Elsevier]

Abstract: The characterization of all additive generators of border-continuous triangular norms (conorms) is introduced. It is also shown that the pseudo-inverse of the Cantor function is an additive generator with a dense set of all points of discontinuity in yielding a border-continuous triangular conorm. [Copyright &y& Elsevier]

Abstract  This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0–1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion. [ABSTRACT FROM AUTHOR]

Abstract: The convex combination of continuous triangular norms is discussed. Let be the class of all nilpotent -norms, be the class of all strict -norms and be the class of all continuous non-Archimedean -norms. It is shown that for any -norms which are taken from two different classes of , the convex combination is not associative for any . The case of two continuous non-Archimedean -norms is also investigated. By our results, the problem of whether or not two continuous -norms can be combined with each other is reduced to that of whether two strict (or two nilpotent) -norms can be combined with each other or not. [Copyright &y& Elsevier]

Abstract  Semi-divisibility of left-continuous triangular norms is a weakening of the divisibility (i.e., continuity) axiom for t-norms. In this contribution we focus on the class of semi-divisible t-norms and show the following properties: Each semi-divisible t-norm with Ran(n T ) = [0, 1] is nilpotent. Semi-divisibility of an ordinal sum t-norm is determined by the corresponding property of its first component (which can be a proper t-subnorm, too). Finally, negations with finite range derived from semi-divisible t-norms are studied. [ABSTRACT FROM AUTHOR]

Semi-divisibility of left-continuous triangular norms is a weakened version of the divisibility axiom. The t-norms on which fulfil the semi-divisibility axiom are studied. The only semi-divisible t-norm with is shown to correspond to the Łukasiewicz t-norm. Several examples of semi-divisible discrete t-norms are shown. [Copyright &y& Elsevier]

In this paper we present some properties of discontinuous reversible triangular norms. Two of them seem to be important: (1) a reversible t-norm T which possesses only trivial idempotent elements must be Archimedean. (2) If T is a discontinuous reversible t-norm and has non-trivial idempotent elements, then T is an ordinal sum of two t-noms, the one on the lower part of the unit square dominating Lukasiewcz t-norm, the other being reversible and Archimedean. Consequently, no left-continuous t-norm that is not continuous is reversible. Therefore, the question of reversibility of discontinuous t-norms is reducible to the case when the t-norms in question possess trivial idempotent elements only. Moreover, we have completely characterized the reversibility of (the reverse of a reversible t-norm T). [Copyright &y& Elsevier]

Abstract: Contour lines totally fix the structure of a t-norm. Expressing the characteristic properties of a left-continuous t-norm in terms of contour lines, provides us more insight into its geometrical structure. In particular, we focus on the decomposition of t-norms that have a continuous contour line. [Copyright &y& Elsevier]

Some topological properties of k-Lipschitz triangular norms are investigated, generalizing the results known for the case [image omitted] . For any fixed [image omitted]  the class of all k-Lipschitz t-norms is shown to be compact. Moreover, it is shown that the set of all k-Lipschitz t-norms is the closure of both the set of all strict k-Lipschitz t-norms and the set of all nilpotent k-Lipschitz t-norms. [ABSTRACT FROM AUTHOR]

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh's extension principle based on a t-norm T. A t-norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation, if this arithmetic operation is closed for this class. It is important to know which t-norms are consistent with a particular type of fuzzy intervals. Recently, Dombi and Győrbíró proved that addition is closed if the Dombi t-norm is used with two bell-shaped fuzzy intervals. A result proved by Mesiar on a strict t-norm based shape preserving additions of LR-fuzzy intervals with unbounded support is recalled. As applications, we define a broader class of bell-shaped fuzzy intervals. Then we study t-norms which are consistent with these particular types of fuzzy intervals. Dombi and Győrbíró''s results are special cases of the results described in this paper. [Copyright &y& Elsevier]

This article gives a new approach to aggregating assuming that there is an indistinguishability operator or similarity defined on the universe of discourse. The very simple idea is that when we want to aggregate two values a and b we are looking for a value λ that is as similar to a as to b or, in a more logical language, the degrees of equivalence of λ with a and b must coincide. Interesting aggregation operators on the unit interval are obtained from natural indistinguishability operators associated to t-norms that are ordinal sums. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 857–873, 2006. [ABSTRACT FROM AUTHOR]

Abstract: Ordinal sums have been introduced in many different contexts, e.g., for posets, semigroups, t-norms, copulas, aggregation operators, or quite recently for hoops. In this contribution, we focus on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets. Necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t-norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t-norms. By such also the structure of the underlying bounded lattice is investigated. Further, it is shown that up to trivial cases there are no ordinal sum t-norms on product lattices in general. [Copyright &y& Elsevier]

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In the literature there are different definitions of conditional possibility. Starting from a general axiomatic definition, we propose a definition of independence for ⊙-conditional possibility, in the case that ⊙ is a strictly monotone triangular norm. We study its main properties to compare it to other definitions introduced in possibility theory. Then, we show that the controversial aspects related to logical dependencies (structural zeros) can be circumvented. Moreover, a set of properties (the well-known graphoid properties) has been considered to be tested, allowing us to compare the proposed definition to the independence notions given in the context of other uncertainty formalisms. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 299–323, 2006. [ABSTRACT FROM AUTHOR]

For 1 < p ≤ 2, an L p -gradient estimate for a symmetric Markov semigroup is derived in a general framework, i. e. $$ {\left\| {\Gamma ^{{1/2}} {\left( {T_{t} f} \right)}} \right\|}_{p} \leqslant \frac{{C_{p} }} {{{\sqrt t }}}{\left\| f \right\|}_{p} $$ , where Γ is a carré du champ operator. As a simple application we prove that Γ1/2(( I- L)-α) is a bounded operator from L p to L p provided that 1 < p < 2 and $$ \frac{1} {2} < \alpha < 1 $$ . For any 1 < p < 2, q > 2 and $$ \frac{1} {2} < \alpha < 1 $$ , there exist two positive constants c q,α, C p,α such that ∥ Df∥ p ≤ C p,α∥( I - L)α f∥ p , c q,α∥( I - L)1-α f∥ q ≤ ∥ Df∥ q + ∥ f∥ q, where D is the Malliavin gradient ([2]) and L the Ornstein–Uhlenbeck operator. [ABSTRACT FROM AUTHOR]