Extensionality with respect to indistinguishability operators

Extensionality is explored form different points of view. Extensional fuzzy subsets from a fuzzy equivalence relation E are considered as observable subsets with respect to the granularity generated by E. Interestingly, they are characterized as the fuzzy subsets that can be obtained as combinations of the fuzzy equivalence classes of E. Extensional mappings are characterized topologically and the set of extensional mappings between two universes are algebraically determined. Specifying the results to fuzzy mappings from a universe X onto [0, 1] an interpretation of type-2 fuzzy subsets of X as fuzzification of its type-1 fuzzy subsets is provided.Peer ReviewedPostprint (author's final draft

Aggregating fuzzy subgroups and T-vague groups

Fuzzy subgroups and T-vague groups are interesting fuzzy algebraic structures that have been widely studied. While fuzzy subgroups fuzzify the concept of crisp subgroup, T-vague groups can be identified with quotient groups of a group by a normal fuzzy subgroup and there is a close relation between both structures and T-indistinguishability operators (fuzzy equivalence relations). In this paper the functions that aggregate fuzzy subgroups and T-vague groups will be studied. The functions aggregating T-indistinguishability operators have been characterized [9] and the main result of this paper is that the functions aggregating T-indistinguishability operators coincide with the ones that aggregate fuzzy subgroups and T-vague groups. In particular, quasi-arithmetic means and some OWA operators aggregate them if the t-norm is continuous Archimedean.Peer ReviewedPostprint (author's final draft

Aggregation operators and lipschitzian conditions

Lipschitzian aggregation operators with respect to the natural T - indistin- guishability operator Et and their powers, and with respect to the residuation ! T with respect to a t-norm T and its powers are studied. A t-norm T is proved to be E T -Lipschitzian and -Lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an Archimedean t-norm T with additive generator t , the quasi- arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to TPeer Reviewe

ET-Lipschitzian aggregation operators

Lipschitzian and kernel aggregation operators with respect to the natural Tindistinguishability operator ET and their powers are studied. A t-norm T is proved to be ET -lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the more stable aggregation operator with respect to T.Peer ReviewedPostprint (published version

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 /

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems