Narrow Results

In this paper we study some modifications and generalizations of additivity in the case of aggregation functions on the unit interval [0, 1]. We prove that some common properties of aggregation functions can be viewed as modifications of additivity — additivity with constraints.We also generalize additivity into pseudo-additivity and k-pseudo-additivity, and provide characterization of pseudo-additive and k-pseudo-additive aggregation functions, including maxitive and k-maxitive aggregation functions.

OWA operators and related aggregation techniques generally focus on input vector with a linear ordering. However, in commonly faced multi-criteria and multi-sources evaluation and decision making, the inputs involved form an evaluation matrix. Considering the fact that the data under evaluation are all with two dimensional meanings, this study explores and proposes four novel preference involved aggregation techniques by using RIM quantifiers and OWA operators. The first two models are both with two steps to carry out the aggregation processes, with one using two times of OWA operators and another considering evaluation matrix as a vector lattice. The last two models come from a whole perspective to direct the aggregation processes, with one arising from a global magnitude view and another based on staggered ordering using two specially defined collections of permutations. Illustrative examples and remarks are also spotted immediately following the proposed models or at suitable positions.

We consider a generalization of aggregation and fusion functions when some of their inputs or outputs are undefined. For simplicity, we represent undefined values by a single NaN (not-a-number) value. We define four main methods of treating undefined values, observe some of their properties, and compare them with several mainstream implementations and norms for handling NaN values. Finally, as a case study demonstrating the apparatus's usefulness, the formalism is applied to the discrete fuzzy transform technique. The formalism allows us to prove a generalized approximation theorem for fuzzy transforms, which removes previously required restrictions on fuzzy partitions for sparse input data.

In this paper we deal with the idempotency equation H(x,x)=x for all x∈[0,1]. In particular we solve it for two special cases. First when H is a convex linear combination of a strict t-norm and its (1-j)-dual and second, when H is a convex linear combination of a special kind of aggregation functions F=<(f,N)> and its N-dual, being these aggregation functions, called L-representable aggregation functions, a kind of functions verifying a similar representation theorem to the classical representation theorem for non strict Archimedean t-norms.

This paper looks over a class of operators introduced in ([2]), called t–operators. Introduced in order to be applied to fuzzy preorders, their properties lead them to be also appropriate in some fields like aggregation problems and expert systems. We characterize these operators as a special combination of a t-norm and a t-conorm on [0, 1] in a similar way of uninorms in ([5]). We study duality and self duality on t–operators with respect to a strong negation N. We also give a classification of continuous t–operators through ordinal sums. Finally, we obtain from some t–operators (those idempotent) a special kind of E.A.F. by extending them to E=∪_n≥1[0,1]ⁿ.

We present an application of lattice theory to the framework of influence in social networks. The contribution of the paper is not to derive new results, but to synthesize our existing results on lattices and influence. We consider a two-action model of influence in a social network in which agents have to make their yes–no decision on a certain issue. Every agent is preliminarily inclined to say either "yes" or "no", but due to influence by others, the agent's decision may be different from his original inclination. We discuss the relation between two central concepts of this model: Influence function and follower function. The structure of the set of all influence functions that lead to a given follower function appears to be a distributive lattice. We also consider a dynamic model of influence based on aggregation functions and present a general analysis of convergence in the model. Possible terminal classes to which the process of influence may converge are terminal states (the consensus states and nontrivial states), cyclic terminal classes and unions of Boolean lattices.

The concept of aggregation within fuzzy sets has been deeply modified from the initial view as mappings where a fixed number of aggregated items, and the aggregation itself, take values in the unit interval to assume a family of mapping and impose certain restrictions to assure consistency and computability, till focus on aggregation as algorithms on lists of objects. This chapter points out how new concepts such as recursivity, map reducibility or hierarchy can be translated into computable aggregations.

We present here a generalized approach for handling null queries that contain compound fuzzy attributes. Null queries are queries that elicit a null answer from the database. Compound fuzzy attributes are ambiguous attributes that are not defined in the original database schema but can be derived from multiple rigid attributes in schema. Compound fuzzy attributes derived from simple numbers were studied by Nomura [11]. In this work, we extend compound fuzzy attributes so that they can be derived from numbers, interval values, scalars, and sets of all these data types. Database management systems that can handle this type of ambiguous attributes in null queries not only can reduce the occurrences of null answers but also provide an improved user-friendly query environment.

This paper introduces a generalization of grouping functions by extending one of the boundary conditions of its definition. More specifically, instead of requiring that “the considered function is equal to one if and only if some of the inputs is equal to one”, we allow the range in which some t-conorm is 1. We call such generalization by a s-grouping function with respect to such t-conorm. Then we analyze the main properties of s-grouping function and introduce some construction methods. Considering that the grouping functions depend only on the t-conorm, it is proposed in this work to extend this condition to any t-conorm.