Archimedean T-norm Research Articles

Continuous generated associative aggregation operators

Continuous generated associative aggregation operators are discussed. The only asymmetric solutions are the trivial operators of the projection to the first or the last coordinate, respectively. The only nontrivial convenient aggregation operators are necessarily symmetric and they are either continuous Archimedean t-norms and t-conorms, or generated nullnorms. The latter may combine any nilpotent t-conorm and any t-norm with any prescribed annihilator k∈ ]0,1[ .

Limit properties of quasi-arithmetic means

Limit properties of the class { M g λ } λ∈(0,∞) of all quasi-arithmetic means generated by λ-powers of a given generator g are studied. Special types of generators of quasi-arithmetic means that uniquely correspond to the additive generators of continuous Archimedean t-norms or t-conorms are considered. It is shown that for λ → ∞ , the situation is similar to that for t-norms and t-conorms [6]. For λ → 0 + , the limit operators are quasi-geometric means. Finally, the limit properties of the class { M g α } α∈(0,∞) of all quasi-arithmetic means generated by functions g α , g α ( x)= g( x α ) are investigated.

Computational aspects of fuzzy arithmetics based on Archimedean t-norms

For fuzzy multiplication and division of (L,R)-numbers we derive approximation formulas by including the exact results using (L,R)-numbers. This permits an error control and meets user's ideas on tractable solutions better than the level sets approach does. As potential application the resolution of linear systems is considered.

Continuous Archimedean t-norms and their bounds

Upper and lower bounds in the class of continuous Archimedean t-norms are studied. The existence of strict (nilpotent) bounds of finite families of strict (nilpotent) t-norms is shown in a constructive way, based on the additive generators of the corresponding t-norms. In general, a nilpotent lower bound and a strict upper bound of a finite system of any continuous Archimedean t-norms can be found. Some examples and some applications are given. By the duality, similar results for bounds of continuous Archimedean t-conorms can be derived.

Resolution of composite fuzzy relation equations based on Archimedean triangular norms

Lately, the sup-t-norm composition of fuzzy relations has been used instead of the well-known max–min. Thus, there is a need for methods of studying and solving sup-t-norm fuzzy relation equations ( t is any t-norm). In this paper, the solution existence problem is first studied and solvability criteria for composite fuzzy relation equations of any t-norm are given. Then, a methodology for solving fuzzy relation equations based on sup- t composition, where t is an Archimedean t-norm, is proposed. This resolution method is simpler and faster than those proposed for covering all the continuous t-norms. The result is important, since, as is shown in the paper, the only continuous t-norm that is not Archimedean is the “minimum”.

Transitive closure and betweenness relations

Indistinguishability operators fuzzify the concept of equivalence relation and have been proved a useful tool in theoretical studies as well as in different applications such as fuzzy control or approximate reasoning. One interesting problem is their construction. There are different ways depending on how the data are given and on their future use. In this paper, the length of an indistinguishability operator is defined and it is used to relate its generation via max- T product and via the representation theorem when T is an Archimedean t-norm. The link is obtained taking into account that indistinguishability operators generate betweenness relations. The study is also extended to decomposable operators.

The law of large numbers for fuzzy numbers with unbounded supports

We study laws of large numbers for mutually T-related fuzzy numbers with unbounded supports where T is an Archimedean t-norm and generalize earlier result of Badard.

On the law of large numbers for mutually [formula omitted]-related [formula omitted]- [formula omitted] fuzzy numbers

We study a law of large numbers for sequences of mutually T-related L- R fuzzy numbers where T is an Archimedean t-norm and the diameter of the support of the fuzzy numbers is not uniformly bounded. Earlier results of Fullér (1992) and Triesch (1993) in this area are extended. Some conditions concerning the shapes of f∘R and f∘L and spreads of L- R fuzzy numbers are given, where f is the additive generator of the Archimedean t-norm T. Several examples are given to illustrate the results.

Necessity Measures and Parametric Inclusion Relations of Fuzzy Sets

A necessity measure N is defined by an implication function. However, specification of an implication function is difficult. Necessity measures are closely related to inclusion relations. In this paper, we propose an approach to necessity measure specification by giving a parametric inclusion relation between fuzzy sets A and B which is equivalent to N A(B) ≥ h. It is shown that, in such a way, we can specify a necessity measure, i.e., an implication function. Moreover, when a necessity measure or equivalently, an implication function is given, then the derivation of an associated parametric inclusion relation is discussed. The associated parametric inclusion relation cannot be obtained for any implication function but only for implication functions which satisfy certain conditions. Applying our results to necessity measures defined by S-, R- and reciprocal R-implications with continuous Archimedean t-norms, associated parametric inclusion relations are shown.

THE LENGTH AND BETWEENNESS RELATIONS OF INDISTINGUISHABILITY OPERATORS

The most common ways used to generate indistinguishability operators, namely as transitive closure of reflexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, are related for archimedean t-norms introducing the notion of length of indistinguishability operators.

Fibred triangular norms

In this paper, we deal with the formula g[ T( f( x), f( y))] where f, g are functions mapping from the unit interval to itself and T is a triangular norm. We are interested in the conditions under which this formula yields a triangular norm. The case, when f is continuous is investigated in detail. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. These new triangular norms are the so-called ‘fibred triangular norms’ which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. We investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, some examples are given for the case when f is not continuous.

A note on law of large numbers for fuzzy numbers in a Banach space

In this paper, we study a law of large numbers for sequences of mutually T-related fuzzy numbers in Banach space where T is an Archimedean t-norm and generalize earlier results of Fuller [Fuzzy Sets and Systems 45 (1992) 299], Triesch [Fuzzy Sets and Systems 58 (1993) 339] and Hong and Kim [Fuzzy Sets and Systems 77 (1996) 349].

Strict Archimedean t-norms and t-conorms as universal approximators

In knowledge representation, when we have to use logical connectives, various continuous t-norms and t-conorms are used. In this paper, we show that every continuous t-norm and t-conorm can be approximated, to an arbitrary degree of accuracy, by a strict Archimedean t-norm ( t-conorm).

A T-sum bound of LR-fuzzy numbers

In this paper, we provide an upper bound and a lower bound of T-sum of LR-fuzzy numbers with different spreads where T is Archimedean t-norm, and also show in three examples how close they are to actual membership functions Furthermore, we study when the membership function of T-sum achieves the upper bound.

Triangular-norm-based addition of fuzzy intervals

Triangular-norm-based addition of fuzzy intervals on the extended real line is reduced to the addition of shapes on [0, ∞]. The addition of fuzzy intervals, including LR-fuzzy intervals, LR-fuzzy numbers, LT- and TR-fuzzy numbers, based on limit t-norms is recalled. For continuous Archimedean t-norms, special types of incoming summands allowing the exact output computation are introduced. Several examples are given.

A note to the addition of fuzzy numbers based on a continuous Archimedean T-norm

A simple method of computing the T-sum of special types of fuzzy numbers is introduced. For the Lukasiewicz t-norm T L the proposed method can be applied to the addition of fuzzy numbers with convex membership functions. The classes of fuzzy numbers leading to ⊕ T - idempotents are studied for continuous t-norms.

On Some Classes of Idempotent Operators

In this paper we deal with the idempotency equation H(x,x)=x for all x∈[0,1]. In particular we solve it for two special cases. First when H is a convex linear combination of a strict t-norm and its (1-j)-dual and second, when H is a convex linear combination of a special kind of aggregation functions F=<(f,N)> and its N-dual, being these aggregation functions, called L-representable aggregation functions, a kind of functions verifying a similar representation theorem to the classical representation theorem for non strict Archimedean t-norms.

The convergence of T-sum of fuzzy numbers on Banach spaces

This paper presents the membership function of finite (or infinite) sum (defined by the sup- t-norm convolution) of fuzzy numbers on Banach spaces, in the case of Archimedean t-norm having convex additive generator function and fuzzy numbers with concave shape function, which generalizes Hong and Hwang's results [1] of the real case. As applications, we calculate the membership function of the limit distribution of Yager's, Hamacher's and Dombi's sum.

A characterization of the ordering of continuous t-norms

Given two t-norms T 1 and T 2, it is quite often difficult or even impossible to check directly whether T 1 ⪕ T 2 . In Schweizer and Sklar (1983), a necessary and sufficient condition is given for continuous Archimedean t-norms to be comparable. We simplify this result, proving a sufficient condition which involves only one argument of a one-place function rather than two arguments. This result is applied to show the monotonicity of some well-known classes of t-norms. Next, we generalize the result of Schweizer and Sklar to the case of all continuous t-norms and present also a sufficient condition, which is again easier to check.

DeMorgan systems on the unit interval

Logical connectives on fuzzy sets arise from those on the unit interval. The basic theory of these connectives is cast in an algebraic spirit with an emphasis on equivalence between the various systems that arise. Special attention is given to DeMorgan systems with strict Archimedean t-norms and strong negations. A typical result is that any DeMorgan system with strict t-norm and strong negation is isomorphic to one whose t-norm is multiplication.