Related papers: On the Existence of $t_r$-Norm and $t_r$-Conorm no…
This paper proves that a binary operation ${\star}$ on ${[0, 1]}$, ensuring that the binary operation ${\curlywedge}$ is a ${t}$-norm or ${\curlyvee}$ is a ${t}$-conorm, is a ${t}$-norm, where ${\curlywedge}$ and ${\curlyvee}$ are special…
This paper studies t-norms on the space $\mathbf{L}$ of all normal and convex fuzzy truth values. We first prove that the only non-convolution form type-2 t-norm constructed by Wu et al. satisfies the distributivity law for meet-convolution…
The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm.…
Uninorms with continuous underlying t-norm and t-conorm are discussed and properties of the set of discontinuity points of such a uninorm are shown. This set is proved to be a subset of the graph of a special symmetric, surjective,…
The uninorms with continuous underlying t-norm and t-conorm are characterized via an extended ordinal sum construction. Using the results of [18], where each uninorm with continuous underlying operations was characterized by properties of…
In this paper, we provide some structures of uninorms on bounded lattices via t-conorms, closure operators and t-subnorms, subject to certain constraints on the closure operators and t-subnorms. Importantly, these constraints are shown to…
Let $T^*:[0,1]^2\rightarrow[0,1]$ be a continuous, non-decreasing and associative function with neutral element, $f: [0,1]\rightarrow [0,1]$ be a strictly monotone function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ be the pseudo-inverse of $f$.…
Although algebraic structures are frequently analyzed using unary and binary operations, they can also be effectively defined and unified through ternary operations. In this context, we introduce structures that contain two constants and a…
An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality…
This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f:…
It is known that the function $f(e^x)/g(e^x)$ is positive definite for some functions $f,g$ implies the operator norm inequality related to $f,g$. We treat functions which have the following form: $f(t) = t^{(1-\sum_{i=1}^n…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
This paper resolves an open problem posed by Schweizer and Sklar in 1983. We establish that the binary operation $\tauTL$ is a triangle function on $\Delp$ if and only if the following three conditions hold: (a) $L$ is a continuous t-conorm…
In this paper, we introduce the notion of nullnorms on bounded trellises and study some basic properties. Based on the existence of $t$-norms and $t$-conorms on arbitrary bounded trellises, we propose some construction methods of nullnorms…
We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_{\psi},$$…
We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now…
We consider the problem of jointly minimizing forms of two Boolean functions $f, g \colon \{0,1\}^J \to \{0,1\}$ such that $f + g \leq 1$ and so as to separate disjoint sets $A \cup B \subseteq \{0,1\}^J$ such that $f(A) = \{1\}$ and $g(B)…
Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^\gamma (t^2-\|x\|^2)^{\mu-\frac12}$ on…
Motivated by a problem in local differential geometry of Cauchy--Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or,…
Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n…