On continuous triangular norms

The problem of converging to any given continuous triangular-norm by a sequence of continuous Archimedean triangular-norms is studied. Such a sequence is built up in a constructive way. The class of well-founded triangular-norms is introduced. It is proved that a sequence of triangular-norms converges to a given continuous triangular-norm if and only if this convergence is uniform. Finally, we prove that every continuous triangular norm is a uniform limit of continuous Archimedean triangular-norms.     

On extensions of triangular norms on bounded lattices

Smallest and largest possible extensions of triangular norms on bounded lattices are discussed. As such ordinal and horizontal sum like constructions for t-norms on bounded lattices are investigated. Necessary and sufficient conditions for the lattice guaranteeing that the extension is again a t-norm are revealed.     

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.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).     

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] ⁿ transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ _μ and ρ _ν . It is shown that if there exist positive constants ? and M such that ? < ρ _ν < M, ? < ρ _ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)^?1 (α + β)^?1 > 1 and ρ _μ ∈ W ^1,α (Θ), ρ _ν ∈ W ^1,β ) (Θ), where W ^1,q denotes a Sobolev class, then the mapping T belongs to the class W ^1,p (Θ).     

Constructions of quotient algebras via triangular norms

In this paper, properties of certain real-valued mappings Δ on binary systems X which satisfy versions of the triangle inequality are investigated. For example, via a quotient construction using Ker Δ = {x: Δ(x) = 0} it is shown that X/Ker Δ is a d-algebra if X is a d-algebra. In addition fuzzy versions of these triangular norms and their properties are considered as well. Finally boundedness conditions on Δ and a concept of magnitude are both introduced and some consequences are derived.     

On the continuity of residuals of triangular norms

In this work a long-standing problem related to the continuity of R-implications, i.e., implications obtained as the residuum of t-norms, has been solved. A complete characterization of the class of continuous R-implications obtained from any arbitrary t-norm is given. In particular, it is shown that an R-implication I_T is continuous if and only if T is a nilpotent t-norm. Using this result, the exact intersection between the continuous subsets of R-implications and (S,N)-implications has been determined, by showing that the only continuous (S,N)-implication that is also an R-implication obtained from any t-norm, not necessarily left-continuous, is the ?ukasiewicz implication up to an isomorphism.     

Construction of Fuzzy σ-algebras using triangular norms

Generalizing the definitions given by the author [Fuzzy Sets and Systems4 (1980), 83–93] we introduce and study T-fuzzy σ-algebras, T being any triangular norm. The main result is that for a large class of triangular norms each T-fuzzy σ-algebra is generated, i.e., consists of all functions μ:X → [0, 1] being measurable with respect to some σ-algebra on X.     

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”.     

Residuated logics based on strict triangular norms with an involutive negation

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)     

On fuzzy subgroups and fuzzy ideals under triangular norms

In this communication, we explicitly point out that the principal results of Liu 1982 basically deduced from the definition of binary operation ° on the set F(X) of all fuzzy subsets of X, also hold if one uses the weaker definition of product under triangular norm °_t. Fuzzy ideals with respect to the triangular norms are also defined.     

Sufficient triangular norms in many-valued logics with standard negation

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms.     

Characterization of fuzzy measures constructed by means of triangular norms

Following the ideas presented by the author (E. P. Klement, J. Math. Anal. Appl.85 (1982), 543–565) finite T-fuzzy measures are introduced, T being a measurable triangular norm. We show that a T-fuzzy measure is always a fuzzy measure, as considered earlier (E. P. Klement, J. Math. Anal. Appl.25 (1980), 330–339). Then we study the relation to the integral with respect to some classical measure. Finally, for some special triangular norms T, we give precise characterizations of the corresponding classes of T-fuzzy measures.     

Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,M,B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map Φ on U is centralized at zero point (i.e., Φ(A)B = AΦ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let δ : U → U be an additive map. It is also shown that the following four conditions are equivalent: (1) δ is specially generalized derivable at zero point, i.e., δ(AB) = δ(A)B + Aδ(B) Aδ(I)B whenever AB = 0; (2) δ is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1 (B) = τ2 (A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space.     

The lattice-theoretic structure of the sets of triangular norms and semi-copulas

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.     

The coextension of commutative pomonoids and its application to triangular norms

Group coextensions of monoids, which generalise Schreier-type extensions of groups, have originally been defined by P.A. Grillet and J. Leech. The present paper deals with pomonoids, that is, monoids that are endowed with a compatible partial order. Following the lines of the unordered case, we define pogroup coextensions of pomonoids. We furthermore generalise the construction to the case that pomonoids instead of pogroups are used as the extending structures.

The intended application lies in fuzzy logic, where triangular norms are those binary operations that are commonly used to interpret the conjunction. We present conditions under which the coextension of a finite totally ordered monoid leads to a triangular norm. Triangular norms of a certain type can therefore be classified on the basis of the presented results.