Aggregation of fuzzy vector spaces

summary:This paper contributes to the ongoing investigation of aggregating algebraic structures, with a particular focus on the aggregation of fuzzy vector spaces. The article is structured into three distinct parts, each addressing a specific aspect of the aggregation process. The first part of the paper explores the self-aggregation of fuzzy vector subspaces. It delves into the intricacies of combining and consolidating fuzzy vector subspaces to obtain a coherent and comprehensive outcome. The second part of the paper centers around the aggregation of similar fuzzy vector subspaces, specifically those belonging to the same equivalence class. This section scrutinizes the challenges and considerations involved in aggregating fuzzy vector subspaces with shared characteristics. The third part of the paper takes a broad perspective, providing an analysis of the aggregation problem of fuzzy vector subspaces from a general standpoint. It examines the fundamental issues, principles, and implications associated with aggregating fuzzy vector subspaces in a comprehensive manner. By elucidating these three key aspects, this paper contributes to the advancement of knowledge in the field of aggregation of algebraic structures, shedding light on the specific domain of fuzzy vector spaces

Crisp and fuzzy motif and arrangement symmetries in composite geometric figures

AbstractThe notions of motif and arrangement symmetries within composite geometric figures are defined. The relationships between these types of symmetry and the symmetry of the whole figure are clarified by making use of the crystallographic concepts of site symmetry and direction symmetry. From this, it has been deduced that a figure with arbitrary symmetry can be assembled from motifs of likewise arbitrary symmetries. If a motif with symmetry GM is placed on a site having the site symmetry GS ⊆ GM, its contribution to the figure symmetry G is only a subgroup G*MO of its direction symmetry GMO where GS = G*MO ⊆ GMO ⊆ GM. Supernumerary symmetry elements of the motif give rise to intermediate or latent symmetries of the figure. A consequent decomposition of a geometric figure into its constituent points reveals that a large part of the O(n) symmetry of every single point is lost when assembling these points to build up the figure. All “lost” symmetries can, however, be detected as intermediate symmetries of this figure. They can be displayed as fuzzy symmetry landscapes and symmetry profiles for a given figure showing all crisp and intermediate symmetries of interest

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Some illustrative examples of permutability of fuzzy operators and fuzzy relations

Composition of fuzzy operators often appears and it is natural to ask when the order of composition does not change the result. In previous papers, we characterized permutability in the case of fuzzy consequence operators and fuzzy interior operators. We also showed the connection between the permutability of the fuzzy relations and the permutability of their induced fuzzy operators. In this work we present some examples of permutability and non permutability of fuzzy operators and fuzzy relations in order to illustrate these results.Postprint (published version

The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications

The Basic Principles of Uncertain Information Fusion. An organized review of merging rules in different representation frameworks

We propose and advocate basic principles for the fusion of incomplete or uncertain information items, that should apply regardless of the formalism adopted for representing pieces of information coming from several sources. This formalism can be based on sets, logic, partial orders, possibility theory, belief functions or imprecise probabilities. We propose a general notion of information item representing incomplete or uncertain information about the values of an entity of interest. It is supposed to rank such values in terms of relative plausibility, and explicitly point out impossible values. Basic issues affecting the results of the fusion process, such as relative information content and consistency of information items, as well as their mutual consistency, are discussed. For each representation setting, we present fusion rules that obey our principles, and compare them to postulates specific to the representation proposed in the past. In the crudest (Boolean) representation setting (using a set of possible values), we show that the understanding of the set in terms of most plausible values, or in terms of non-impossible ones matters for choosing a relevant fusion rule. Especially, in the latter case our principles justify the method of maximal consistent subsets, while the former is related to the fusion of logical bases. Then we consider several formal settings for incomplete or uncertain information items, where our postulates are instantiated: plausibility orderings, qualitative and quantitative possibility distributions, belief functions and convex sets of probabilities. The aim of this paper is to provide a unified picture of fusion rules across various uncertainty representation settings

Aggregation of L-probabilistic quasi-uniformities

[EN] The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to merge an arbitrary family of (bases of) L-probabilistic quasi-uniformities into a single one. These fuzzy structures are special filters of fuzzy binary relations. Hence we first make a complete study of functions between partially-ordered sets that preserve some special sets, such as filters. Afterwards, a complete characterization of those functions aggregating bases of L-probabilistic quasi-uniformities is obtained. In particular, attention is paid to the case L={0,1}, which allows one to obtain results for functions which aggregate crisp quasi-uniformities. Moreover, we provide some examples of our results including one showing that Lowen's functor iota which transforms a probabilistic quasi-uniformity into a crisp quasi-uniformity can be constructed using this aggregation procedure.J. Rodriguez-Lopez acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion Proyecto PGC2018-095709-B-C21.Pedraza Aguilera, T.; Rodríguez López, J. (2020). Aggregation of L-probabilistic quasi-uniformities. Mathematics. 8(11):1-21. https://doi.org/10.3390/math8111980S12181