Category Of Fuzzy Sets Research Articles

Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses

The research presented in this article was conducted on a representative and purposefully selected sample of 102 residential buildings that were erected in the second half of the nineteenth and early twentieth centuries in the downtown district of Wroclaw (Poland). The degree of the technical wear of an old residential building is determined by the conditions of its maintenance and operation. The diagnosis of the impact of the maintenance of residential buildings on the degree of their technical wear was carried out using quantitative methods in the categories of fuzzy sets and also by using the authors’ own models created in fuzzy conditions. It was proved that the expression of the operational state of a building, considered as the process that plays the greatest role in its accelerated destruction, is mechanical damage to the internal structure of its elements. This damage is determined in the categories of fuzzy sets and has a high frequency and a cumulative effect of occurrence, which are characteristic for buildings in satisfactory and average maintenance conditions. The use of simple operations in fuzzy set calculus enabled the impact of elementary damage that occurs with a specific frequency, as well as the measure of its correlation on the observed technical wear of building elements to be considered. As a result, it was possible to identify the elementary damage that determines the degree of the technical wear of a building element. For each of the selected building elements, the maximal and minimal fuzzy relational equations (damage and technical wear) were determined. Their solutions were given in the form of clear relational matrixes that constitute big data arrays. They define the domain and range of the maximal and minimal fuzzy relations, the height of the fuzzy relations, their differences, and the place of their occurrence between the maximal and minimal dependencies.

Finite point sets in recognizing location and orientation of machine parts of complex shapes

In the article, a method of recognizing the position, location and orientation of irregular machine parts with a complex outline of the external contour is suggested. Recognition is performed on the basis of a raster image that has undergone preliminary processing in a vision system. The developed method is based on matching the finite set of points retrieved from the external contour of the classified object. The originality of the approach consists of the form of the object pattern taken into account in determining the degree of similarity to reference objects. The similarity index was defined in the category of fuzzy sets. A cellular automaton has been proposed to generate the outline of the perimeter quickly while simultaneously smoothing it out. The process chain necessary for the recognition function to be implemented has been presented. The developed method has been illustrated with an example related to recognizing the position, location and orientation of a rear wheel pin. The results were compared to the classic Blair-Bliss, Danielsson and Haralick shape coefficients. The sensitivity of the developed method to a change in the scale and rotating the object as well as errors in outlining its edge has been tested. The obtained results confirmed the advantageous features of the developed method, both in the aspect of the recognition quality and the practically viable time of waiting for the results of processing.

Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

Images and Preimages of Subobjects under the Morphisms in a New Category of Fuzzy Sets-I

This paper is the third in a sequence of papers on categories by the same authors. In one of the papers, a new category of fuzzy sets was defined and a few results were established pertaining to that special category of fuzzy sets S. Here, the concept of a fuzzy subset of a fuzzy set is defined under the category S. Besides, the notions of images and preimages of fuzzy sets are also defined under morphisms in the category of fuzzy sets and how smoothly these images and preimages behave under the action of these morphisms is analyzed. Finally, results have been proved on algebra of morphisms of this category S.

A fuzzy number ranking in project selection for cultural heritage sites

Abstract Any conservation and exploitation project of archaeological sites has the managerial aspects of a strategic intervention. Its character is defined by the variety of factors to be considered (durability of the materials and structures, landscape conservation, maintenance of cultural values and economic impact), the complexity of each of them (including those that are strictly technical such as those which relate to the structures and materials) and the projections of these actions into future, the forecast of which is intrinsically uncertain (wars, earthquakes, climate and cultural changes, and kinetics of chemical–physical–biological phenomena). We face the problem of creating a standardised but clear and easily understandable framework for the strategic decision of selection among the diverse projects of conservation and of cultural and economic enhancement. The characters of fuzzy logic and mathematic are extremely interesting in such situations, which are characterised by a quantitative and qualitative combination of factors, inevitably partial knowledge, intrinsic complexity, and the need for projection into the future. We propose the creation of a hierarchy among the different projects based on ratings expressed as fuzzy numbers. Fuzzy numbers are a particular category of fuzzy sets and can be considered an extension of the concept of “ordinary” numbers. Teams of experts on the different factors are called upon to give their respective fuzzy ratings based on scientific knowledge acquired with laboratory, environmental and theoretic investigations as well as on their experience during the implementation, management, and critical assessment of similar projects. Fuzzy numbers mathematics can be the bearing structure of the language and of proposed ranking method. This is a procedure in which the data set of a given archaeological site can be framed rationally and holistically, and therefore one can proceed to the progressive fine tuning of future projects. We shall apply the procedure discussed to select a portfolio of projects of conservation/enhancement for the Roman Amphitheatre at Nora (southern Sardinia). It is one of the most complete and important structures of the site, situated less than 100 m from the sea shore. The state of alteration of the cavea is particularly marked; the blocks of andesitic rock are intensely fractured and affected by consistent detachments, and in places where they are completely absent, the foundation tier, built in opus caementicium is uncovered, and appears affected by deep pulverisation. The anthropic erosion is intense. The fuzzy rating procedure clearly defines the best project between the five proposed.

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches.

Categories of fuzzy sets and weak topos

In this paper, definitions of middle object in a category and concept of weak topos are given. A weak topos is a Cartesian closed category with a middle object. It is proved that category of functors from a small category to category Fuz of fuzzy sets is a weak topos.

Fuzzy Set Theory and Topos Theory

AbstractThe relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is explicated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.

Category of Fuzzy Sets

In the present paper, the concept of category theory, the category of fuzzy sets and the theory of flou sets are introduced. This article also shows the isomorphism between the category of fuzzy sets and flou sets. Keywords Category, flou sets, fuzzy sets, morphism, functor DOI: http://dx.doi.org/10.3126/av.v1i0.5301 Academic Voices 2011; 1: 13-15