On the construction of t-norms (t-conorms) by using interior (closure) operator on bounded lattices

summary:Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karaçal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices

On the constructions of t-norms and t-conorms on some special classes of bounded lattices

summary:Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples

Construction of k-Lipschitz triangular norms and conorms from empirical data

This paper examines the practical construction of k-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on k-convex additive generators and translate k-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees k-Lipschitz property of the resulting triangular norms and conorms.<br /

Edge detection on DICOM image using triangular norms in Type-2 fuzzy

In image processing, edge detection is an important venture. Fuzzy logic plays a vital role in image processing to deal with lacking in quality of an image or imprecise in nature. This present study contributes an authentic method of fuzzy edge detection through image segmentation. Gradient of the image is done by triangular norms to extract the information. Triangular norms (T norms) and triangular conorms (T conorms) are specialized in dealing uncertainty. Therefore triangular norms are chosen with minimum and maximum operators for the purpose of morphological operations. Also, mathematical properties of aggregation operator to represent the role of morphological operations using Triangular Interval Type-2 Fuzzy Yager Weighted Geometric (TIT2FYWG) and Triangular Interval Type-2 Fuzzy Yager Weighted Arithmetic (TIT2FYWA) operators are derived. These properties represent the components of image processing. Here Edge detection is done for DICOM image by converting into 2D gray scale image, using Type-2 fuzzy MATLAB and which is the novelty of this work

Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

About t-norms on type-2 fuzzy sets.

Walker et al. defined two families of binary operations on M (set of functions of [0,1] in [0,1]), and they determined that, under certain conditions, those operations are t-norms (triangular norm) or t-conorms on L (all the normal and convex functions of M). We define binary operations on M, more general than those given by Walker et al., and we study many properties of these general operations that allow us to deduce new t-norms and t-conorms on both L, and M