On the distributivity equation for uni-nullnorms

summary:A uni-nullnorm is a special case of 2-uninorms obtained by letting a uninorm and a nullnorm share the same underlying t-conorm. This paper is mainly devoted to solving the distributivity equation between uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms and some binary operators, such as, continuous t-norms, continuous t-conorms, uninorms, and nullnorms. The new results differ from the previous ones about the distributivity in the class of 2-uninorms, which have not yet been fully characterized

An extension of the ordering based on nullnorms

summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms

Migrativity properties of 2-uninorms over semi-t-operators

summary:In this paper, we analyze and characterize all solutions about $\alpha$-migrativity properties of the five subclasses of 2-uninorms, i. e. $C^{k}$, $C^{0}_{k}$, $C^{1}_{k}$, $C^{0}_{1}$, $C^{1}_{0}$, over semi-t-operators. We give the sufficient and necessary conditions that make these $\alpha$-migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for $G\in C^{k}$, the $\alpha$-migrativity of $G$ over a semi-t-operator $F_{\mu,\nu}$ is closely related to the $\alpha$-section of $F_{\mu,\nu}$ or the ordinal sum representation of t-norm and t-conorm corresponding to $F_{\mu,\nu}$. But for the other four categories, the $\alpha$-migrativity over a semi-t-operator $F_{\mu,\nu}$ is fully determined by the $\alpha$-section of $F_{\mu,\nu}$

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

Distributivity of ordinal sum implications over overlap and grouping functions

summary:In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given

Aggregation functions: Means

The two-parts state-of-art overview of aggregation theory summarizes the essential information concerning aggregation issues. Overview of aggregation properties is given, including the basic classification of aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with open arity (extended means).

Fuzzy Mathematics

This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value