Continuous Triangular Norms Research Articles

Principles of inclusion and exclusion for fuzzy sets

Many previous papers have presented generalizations of the principle of inclusion and exclusion to fuzzy sets or IF-sets (Atanassov's intuitionistic fuzzy sets) with Gödel or product operations. We clarify for which fuzzy intersections and unions, based on continuous triangular norms and conorms, the principle of inclusion and exclusion holds.

T-proximity compatible with T-neighbourhood structure

Abstract In this paper, we show that every T -neighbourhood space induces a T -proximity space, where T stands for any continuous triangular norm. An axiom of T -completely regular of T -neighbourhood spaces introduced by Hashem and Morsi (2003) [3] , guided by that axiom we supply a Sierpinski object for category T -PS of T -proximity spaces. Also, we define the degree of functional T -separatedness for a pair of crisp fuzzy subsets of a T -neighbourhood space. Moreover, we define the Cech T -proximity space of a T -completely regular T -neighbourhood space, hence, we establishes it is the finest T -proximity space which induces the given T -neighbourhood space.

Characterization of TL-uniform spaces by coverings

In this paper we introduce the concepts of fuzzy TL -uniform spaces by means of coverings, where T stands for any continuous triangular norm. We show that the structure of covering TL -uniform spaces are isomorphic to fuzzy TL -uniform spaces as defined by Hashem and Morsi (2006) [4] . In particular, we study the continuity of functions between covering TL -uniform spaces, the I -topological space associated with a covering TL -uniform space. Also, we define the notions of level covering uniformities for a covering TL -uniformity. Moreover, we deduce a number of functors between categories of covering TL -uniform spaces, fuzzy TL -uniform spaces, covering uniform spaces and uniform spaces.

Cross-migrative triangular norms

We study the cross-migrativity of triangular norms. The classes of continuous triangular norms, which are cross-migrative with respect to some strict or nilpotent triangular norm, respectively, are completely characterized, as well as those which are cross-migrative with respect to the greatest and smallest triangular norms, respectively. As a by-product, parametric systems of equivalence relations on the classes of strict and nilpotent triangular norms are found. © 2012 Wiley Periodicals, Inc.

Distributive Equations of Implications Based on Continuous Triangular Norms (I)

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this paper, we explore the distributive equations of implications. In detail, by means of the sections of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I$ </tex></formula> , we give out the sufficient and necessary conditions of solutions for the distributive equation of implication <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I(x,T_1(y,z))=T_2(I(x,y),I(x,z))$</tex></formula> , when <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$T_1$</tex></formula> is a continuous but not Archimedean triangular norm, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$T_2$</tex></formula> is a continuous and Archimedean triangular norm, and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I$</tex></formula> is an unknown function. This obtained characterizations indicate that there are no continuous solutions for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I$</tex></formula> is continuous except for the point (0,0), we get its complete characterizations. Here, it should be pointed out that these results make differences with recent results that are obtained by Baczyński and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.

A note on metrics induced by copulas

In this short note, we provide a continuous triangular norm that is not a copula and that generates a metric. Thus, the 9th open problem collected in Alsina et al. [Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006] has a negative answer.

T-Norm based cuts of intuitionistic fuzzy sets

We consider the system of intuitionistic fuzzy sets (IF-sets) in a universe X and study the cuts of an IF-set. Suppose a left continuous triangular norm is given. The t-norm based cut (level set) of an IF-set is defined in a way that binds the membership and nonmembership functions via the triangular norm. This is an extension of usual cuts of IF-sets. We show that the system of these cuts fulfils analogical properties as usual systems of cuts. However, it is not possible to reconstruct an IF-set from the system of t-norm based cuts.

Fuzzy logics with an additional involutive negation

This paper surveys the present state of knowledge on propositional fuzzy logics extending SBL with an additional involutive negation. The involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms.

Solution of an open problem of Alsina, Frank and Schweizer

An open problem of Alsina, Frank and Schweizer on associative functions is solved. Equivalence between continuity in each component and joint continuity of an associative two place function satisfying some boundary conditions is shown, thus providing a minimal set of axioms in order for a two place function to be a continuous triangular norm.

Supermigrative semi-copulas and triangular norms

A semi-copula S : [ 0 , 1 ] 2 → [ 0 , 1 ] is called supermigrative if it is commutative and satisfies S ( α x , y ) ⩾ S ( x , α y ) for all α ∈ [ 0 , 1 ] and for all x , y ∈ [ 0 , 1 ] such that y ⩽ x . In this paper, the class of supermigrative semi-copulas is investigated, by focusing, in particular, on the subclass of continuous triangular norms. Some interesting connections with the theory of copulas are also underlined.

Convex combinations of nilpotent triangular norms

In this paper we deal with the open problem of convex combinations of continuous triangular norms stated by Alsina, Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140, Problems 5 and 6]. They pose a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. The main result of this paper gives a negative answer to the question for any pair of continuous Archimedean triangular norms with different supports. With the help of this result we show that a non-trivial convex combination of nilpotent t-norms is never a t-norm. The main result also gives an alternative proof to the result presented by Ouyang and Fang [Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., 68 (11) (2008) 3382–3387, Theorem 3.1]. In proof of the main theorem we utilize the Reidmeister condition known from the web geometry.

Continuity of triple I methods based on several implications

This paper focuses on the continuity problem of fully implicational triple I methods for fuzzy reasoning. Based on the residual implications generated by continuous triangular norms, the residual implication generated by nilpotent minimum with the standard negation and the Zadeh implication, respectively, continuity and uniform continuity properties of triple I methods are examined in Hamming and uniform metrics. We also investigate the continuity of the model of a fuzzy if-then rule corresponding to the triple I methods.

Some results of weighted quasi-arithmetic mean of continuous triangular norms

The weighted quasi-arithmetic mean of continuous triangular norms is investigated. Some results on convex combination of continuous triangular norms in Ouyang and Fang [Y. Ouyang, J.X. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal. – TMA 68 (2008) 3382–3387] are generalized. Several open problems are posed.

Dominance is not transitive on continuous triangular norms

Dominance is a relation between operations which are defined on a poset. It is well known that dominance of t-norms is reflexive and antisymmetric. The question whether dominance of t-norms is transitive was posed in [5, Problem 12.11.3]. In this paper we show that dominance is not transitive on the set of continuous t-norms.

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

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.

Strong law of large numbers for t-normed arithmetics

We prove a strong law of large numbers for sequences of pairwise i.i.d. fuzzy random variables. Addition is given by the extension principle associated with a general continuous triangular norm. Our result includes the SLLN for levelwise sums, obtained when the chosen triangular norm is the minimum.

Spectral Duality for Finitely Generated Nilpotent Minimum Algebras, with Applications

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Godel-Dummett logic), mutatis mutandis. We give several such applications.

Constructions in the category of fuzzy T-proximity spaces

This paper is devoted to study the initial fuzzy T-proximity structures of the fuzzy T-proximity spaces defined by Hashem and Morsi in 2002, where T stands for any continuous triangular norm. In this paper, we show that all initial and final lifts in the category T-FPS of these fuzzy T-proximity spaces exist and hence the initial and final fuzzy T-proximity structures exist. We introduce a characterization for the initial fuzzy T-proximity structures, so as a special initial fuzzy T-proximity spaces, the subspaces and product spaces of these fuzzy T-proximity spaces can also be characterized. We also show that the fuzzy topology associated to the initial fuzzy T-proximity structure of a family of fuzzy T-proximity spaces, coincides with the initial fuzzy topology of the family of fuzzy topologies associated to these fuzzy T-proximity spaces.

Some observations about the convex combination of continuous triangular norms

The convex combination of continuous triangular norms is discussed. Let C 1 be the class of all nilpotent t -norms, C 2 be the class of all strict t -norms and C 3 be the class of all continuous non-Archimedean t -norms. It is shown that for any t -norms T 1 , T 2 which are taken from two different classes of C i ( i = 1 , 2 , 3 ) , the convex combination T = ( 1 − λ ) T 1 + λ T 2 is not associative for any λ ∈ ( 0 , 1 ) . The case of two continuous non-Archimedean t -norms is also investigated. By our results, the problem of whether or not two continuous t -norms can be combined with each other is reduced to that of whether two strict (or two nilpotent) t -norms can be combined with each other or not.

On continuous triangular norms that are migrative

In this paper we completely describe all continuous migrative triangular norms. Since the migrative property excludes both idempotent and nilpotent classes, the characterization and construction is carried out by solving a functional equation for additive generators of strict t-norms. We also study cases when the construction results in a smooth generator.