We are interested in generalizations of the concepts of t-norm and t-conorm. In a companion chapter in this volume [14] we have presented a family of hypert-norms ∧q and a family of hyper-t-conorms ∨p. The prefix hyper is used to indicate multi-valued operations, also known as hyperoperations (see [5, 14]), i.e. operations which map pairs of elements to sets of elements. See [14] for the constr...

Keywords: mathematical fuzzy logic, basic fuzzy logic, monoidal t-norm logic, hoop logic, non-commutative fuzzy logic. This paper is a companion to my [12]. The latter is a rather technical paper developing a generalized (mathematical) fuzzy logic. Main definitions and results of [12] are reproduced here without any proofs; but this is preceded by a survey on the basic proposi-tional fuzzy logi...

Aggregation processes are fundamental in any discipline where the fusion of information is of vital interest. For aggregating binary fuzzy relations such as equivalence relations or fuzzy orderings, the question arises which aggregation operators preserve specific properties of the underlying relations, e.g. T -transitivity. It will be shown that preservation of T -transitivity is closely relat...

The aim of this paper is to promote web geometry and, especially, the Reidemeister closure condition as a powerful and intuitive tool characterizing associativity of the Archimedean triangular norms. In order to demonstrate its possible applications, we provide the full solution to the problem of convex combinations of nilpotent triangular norms. Keywords— Archimedean triangular norm, web geome...

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, x_n)|=\|T\|,$ where ${\mathcal denotes the space all continuous $n$-linear forms on $E.$For E),$ we define $${Norm}(T)=\Big\{(x_1, E^n: (x_1, x_n)~\mbox{is point of}~T\Big\}.$$${Norm}(T)$ set} $T$. We classify ${Norm}(T)$ for every L}_s(^3 l_{1}^2)$.

that produce the \best ̄t" to the data. Here, n denotes the noise or ̄tting error. By the term \best ̄t," we mean the minimization of the ̄tting error in some sense. For example, typical norm or norm-like functionals used as arguments in the above minimization problem are: 2 knk1 = ess. supt jn(t)j; the L1 norm. 2 knk2 = ¡R1 0 jn(t)j2 dt¢1=2; the L2 norm (energy). 2 knk2;± = ¡R1 0 e2±tjn(t)j2 dt¢1=...

Using a t-norm T , the notion of T -fuzzy subhypernear-rings (for short TFSrings) of hypernear-rings is introduced and some of their properties are investigated. Also we study the structure of TFS-rings under direct product.

In the paper aggregations of fuzzy relations using functions of n variables are considered. After recalling properties of fuzzy relations, aggregation functions which preserve: reflexivity, irreflexivity, T -asymmetry, T -antisymmetry, symmetry, S-connectedness, T -transitivity, negative Stransitivity, T -S-semitransitivity and T -S-Ferrers property of fuzzy relations, where T is a t-norm and S...

Let X represent either a space C[−1,1] or L α,β(w), 1 ≤ p < ∞, of functions on [−1,1]. It is well known that X are Banach spaces under the sup and the p-norms, respectively. We prove that there exist the best possible normalized Banach subspaces Xα,β of X such that the system of Jacobi polynomials is densely spread on these, or, in other words, each f ∈ Xα,β can be represented by a linear combi...