Fuzzy sets: A topos-logical point of view

Bénabou deduction-categories are defined, with a set of additional assumptions that define categories with formal finite limits (resp. formal regular categories, formal logoi, formal topoi). They are shown to be generalized structures in which higher-order many-sorted languages can be realized. The corresponding Gentzen-type higher-order calculus of sequents is explicited and the soundness theorem is formulated. A construction is given, which associates to each deduction category with formal properties a real category with the corresponding real properties, in a universal way. The corresponding sounddess and completeness properties are formulated for the real categories thus obtained. Fuzzy sets, as generalized by Goguen are introduced, considered as the objects of a category Fuz(H), which turns out to be the real category associated to a very simple formal topos, and thus to be itself a topos: furthermore this is proved to be a Grothendieck topos which is a strictly full epireflective subcategory of Higgs' category of ‘H-valued sets’. Topoi are proposed as generalized fuzzy sets, and deductio0-categories as generalized² fuzzy sets. Some related topics such as Arbib-Manes fuzzy theories, probability, many-valued and fuzzy logics, intensional logic are very briefly touched upon.     

A new approach to multi-criteria sorting based on fuzzy outranking relations: The THESEUS method

In this paper, we propose the THESEUS method, a new approach based on fuzzy outranking relations to multi-criteria sorting problems. Compared with other outranking-based methods, THESEUS is inspired by another view of multi-criteria classification problems. It utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined. This way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects. The appropriate assignment is determined by solving a simple selection problem.The capacity of a reference set for making appropriate assignments is related to a good characterization of the categories. A single reference action characterizing a category may be insufficient to achieve well-determined assignments. In this paper, the reference set capacity to perform appropriate assignments is characterized by some new concepts. This capacity may be increased when more objects are added to the reference set. THESEUS is a method for handling the preference information contained in such larger reference sets.     

Théorie tannakienne non commutative

Inspired by a recent paper by Deligne [2], we extend the Tannaka-Krein duility results (over a field) to the non-commutative situation. To be precise, we establish a 1-1 corresponde:ice between ‘tensorial autonomous categories’ equipped with a ‘fibre functor’ (i. e. tannakian categories without the commutativity condition on the tensor product), and ‘quantum groupoids’ (as defined by Maltsiniotis, [9]) which are ‘transitive’ (7.1.). When the base field is perfect, a quantum groupoid over Spec B is transitive iff it is projective and faithfully fiat over B? _k B. Moreover, the fibre functor is unique up to ‘quantum isomorphism’ (7.6.). Actually, we show Tannaka-Krein duality results in the more general setting where there is no monoidal structure on the category (and the functor); the algebraic object corresponding to such a category is a ‘semi-transitive’ coalgebroid (5.2. and 5.8.).     

The Blackett Memorial Lecture 18 November 1997

It is argued that movement toward the ‘control’ aspects of management should be effected to expand OR/MS activities beyond their present (almost exclusive) emphasis on ‘planning’ functions. This will involve increasing the amount of empirical-inferential approaches to research (based on ex-post data) as one way to help bring this about. Examples of such empirical-inferential research activities in OR/MS are supplied by reference to the early (founding) work of PMS Blackett and others on high-level policy problems during World War II. Extensions to the control function are evidenced in the work of Robert Fetter and others in the development of Diagnostic Related Groups on which the Prospective Payment Systems are based with large and important impacts on the health care delivery systems in the USA as well as other countries. Other examples covered include the use of OR concepts in the early ‘turnaround’ of Federal Express from failure to success. Other new methods will nevertheless be needed to extend and improve presently available OR/MS approaches. Some of these possibilities are illustrated with Data Envelopment Analysis. Ways to combine DEA with commonly used statistical methods are described to show how new and old methods may be combined to further enhance their power and range. Similar relations of DEA to other OR/MS approaches are now being studied, some of which are presently occurring with ‘fuzzy sets’ and ‘multiple objective programming.’     

Fuzzy real algebra: Some results

In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.     

Interpretation of fuzzy data by means of fuzzy rules with applications to speech recognition

Recently, a speech recognition methodology has been proposed which has as one of its main principles the explicit assumption of intrinsic uncertainty of the data (speech signals) and inexactness of the knowledge (acoustic phonetic, etc…) available to interpret them. The main problem presented by this methodology is that of parsing ‘fuzzy data’ by means of ‘fuzzy rules’. To solve this problem, an appropriate fuzzy parsing and interpretation scheme has been proposed. It assumes the data to be represented as strings of ‘fuzzy symbols’, defined as fuzzy sets over the appropriate set of categories, and knowledge as finite-state networks with the arcs labelled by fuzzy symbols of the same type. A formal presentation of this scheme is the main topic of this paper. Included is a brief discussion of the application to Automatic Speech Recognition, and a summary of some results obtained from an implementation example.     

Where do we stand on measures of uncertainty,ambiguity, fuzziness,and the like?

It is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty. Each of these types requires a measure by which the degree of uncertainty of that type can be determined.Two main categories of uncertainty are connected with the terms ‘vagueness’ (or ‘fuzziness’) and ‘ambiguity’. In general, vagueness is associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity, on the other hand, is associated with one-to-many relations, i.e., situations with two or more alternatives that are left unspecified. While the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness, the concept of a fuzzy measure is a general framework for dealing with ambiguity.Several classes of measures of vagueness, usually referred to as measures of fuzziness, have been proposed in the literature. Each class is based on some underlying conception of the degree of fuzziness. A general set of requirements for measures of fuzziness is formulated, followed by an overview of the measures proposed in the literature.Measures of ambiguity are discussed within the framework of plausibility and belief measures. Although it does not cover all fuzzy measures, this framework is sufficiently broad for most practical purposes, and represents a generalization of both probability theory and possibility theory.It is argued that three complementary measures of ambiguity should be employed. One of them is obtained by generalizing the Hartley measure of uncertainty; it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures. The other two are obtained by generalizing the well known Shannon measure of uncertainty; they measure the degree of dissonance and the degree of confusion in evidence, respectively. Basic mathematical properties of these measures are overviewed.It is also argued that each of the four types of uncertainty measures, which are fundamentally different from each other, can be used for measuring structural (syntactic) information in the same sense as the Hartley and Shannon measures have been used in this respect. As such, these measures are potentially powerful tools for dealing with systems problems such as systems modelling, analysis, simplification, or design.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

A multiple criteria sorting method where each category is characterized by several reference actions: The Electre Tri-nC method

This paper presents Electre Tri-nC, a new sorting method which takes into account several reference actions for characterizing each category. This new method gives a particular freedom to the decision maker in the co-construction decision aiding process with the analyst to characterize the set of categories, while there is no constraint for introducing only one reference action as typical of each category like in Electre Tri-C (Almeida-Dias et al., 2010). As in such a sorting method, this new sorting method is composed of two joint rules. Electre Tri-nC also fulfills a certain number of natural requirements. Additional results on the behavior of the new method are also provided in this paper, namely the ones with respect to the addition or removal of the reference actions used for characterizing a certain category. A numerical example illustrates the manner in which Electre Tri-nC can be used by a decision maker. A comparison with some related sorting procedures is presented and it allows to conclude that the new method is appropriate to deal with sorting problems.     

Bar Categories and Star Operations

We introduce the notion of ‘bar category’ by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional *-Hopf algebra and modules over a more general object which we call a ‘quasi-*-Hopf algebra’ and for which examples include the standard quantum groups $u_q(\mathfrak{g})$ at q a root of unity (these are well-known not to be usual *-Hopf algebras). We also provide examples of strictly quasiassociative bar categories, including modules over ‘*-quasiHopf algebras’ and a construction based on finite subgroups H???G of a finite group. Inside a bar category one has natural notions of ‘?-algebra’ and ‘unitary object’ therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and ?-braided groups (Hopf algebras) in braided-bar categories. Examples include the transmutation B(H) of a quasitriangular *-Hopf algebra and the quantum plane ${\mathbb C}_q^2$ at certain roots of unity q in the bar category of $\widetilde{u_q(su_2)}$ -modules. We use our methods to provide a natural quasi-associative C ^*-algebra structure on the octonions ${\mathbb O}$ and on a coset example. In the Appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to *-Hopf algebras.     

On chaotic graphs and applications in physics and biology

El Naschie’s ε^∞ theory in Quantum space time is given and discussed geometrically and topologically as a category of fuzzy spaces, these fuzzy categories in which lines are fuzzy fractal lines. In this paper, we represent the chaotic graphs as many fuzzy fractal lines up to ∞. We will describe them by chaotic matrices. Many fuzzy systems (chaotic systems) are described and applied in [8], [9], [10], [11], [12]. This article introduces some operations on the chaotic graphs such as the union and the intersection; also both of the chaotic incidence matrices and the chaotic adjacency matrices representing the chaotic graphs induced from these operations will be studied. Theorems governing these studies are obtained. Some applications on chaotic graphs are given [18], [19], [20], [21].     

Transportation policies for single and multi-objective transportation problem using fuzzy logic

In this paper, single and multi-objective transportation models are formulated with fuzzy relations under the fuzzy logic. In the single-objective model, objective is to minimize the transportation cost. In this case, the amount of quantities transported from an origin to a destination depends on the corresponding transportation cost and this relation is verbally expressed in an imprecise sense i.e., by the words ‘low’, ‘medium’, ‘high’. For the multi-objective model, objectives are minimization of (i) total transportation cost and (ii) total time for transportation required for the system. Here, also the transported quantity from a source to a destination is determined on the basis of minimum total transportation cost as well as minimum transportation time. These relations are imprecise and stated by verbal words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. Both single objective and multi-objective problems using Real coded Genetic Algorithms (GA and MOGA) are developed and used to solve the single level and bi-level logical relations respectively. The models are illustrated with numerical data and optimum results are presented.     

Dualité tannakienne pour les quasi-groupoïdes quantiques

This paper is the sequel of a previous one [2] where we extended the Tannaka-Krein duality results to the non-commutative situation, i.e. to ‘quantum groupoids’. Here we extend those results to the quasi-monoidal situation, corresponding to ‘quasi-quantum groupoids’ as defined in [3] (‘quasi-’ stands for quasi-associativity a la Drinfeld). More precisely, let B be a commutative algebra over a field k. Given a tensor autonomous category τ,. we define the notion of a quasi-fibre functor ω:τ-proj B (here, ‘quasi-’ means without compatibility to associativity constraints). On the other hand, we define the notion of a transitive quasi-quantum groupoid over B. We then show that the category of tensor autonomous categories equipped with a quasi-fibre functor (with suitable morphisms), is equivalent to the category of transitive quasi-quantum groupoids (5.4.2)

Moreover, we classify quasi-fibre functors for a semisimple tensor autonomous category (6.1.2), and give a few examples : a family of quantum groups having the same tensor category of representations as Sl²(C), but with non-isornorphic underlying coalgebras, constructed by means of an R-matrix introduced by Gurevich ([9]) in a manner suggested to the author by Lyubashenko (6.2.1 and 6.2.2), and quasi-quantum groups which cannot be obtained from quantum groups by a Drinfeld twist (6.2.1)     

Pre-recursive categories

How can we obtain in a natural way the primitive recursive functions in categories? In this paper, we study the free ‘cartesian closed category with a natural numbers object (in the sense of the Peano-Lawvere axiom)’ generated by the empty category. In this category, every morphism 1 → N represents a natural number and every morphism N → N represents a function. Furthermore, the set of functions represented by the morphisms of this category contains strictly the set of primitive recursive functions and is strictly contained in the set of recursive functions. Then, we see that this category is a categorical version of Grzegorczyk's recursive functionals of finite type, with the addition of product types.     

ON THE EXISTENCE AND NON-EXISTENCE OF COMPLEMENTARY RADICAL AND SEMISIMPLE CLASSES

ABSTRACT

In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.     

Fuzzy cardinals based on the generalized equality of fuzzy subsets

The purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality. The proposed idea will be compared with several approaches which were earlier offered in ‘fuzzy’ literature. Using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals, i.e. we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has, respectively, at most/least and less than/more than k elements. Various properties of different fuzzy cardinals and semicardinals, as well as some interdependences between them, are also given.     

Conditioning in fuzzy environment

The concepts of ‘conditional fuzzy algebra’ and ‘conditional evaluation of possibility’ are introduced when the conditioning event is either in the initial algebra or does not belong to it. In the latter case a ‘wider’ fuzzy environment in which ‘old’ and ‘new’ elements could be considered together is constructed, and a new evaluation of possibility which allows to evaluate the elements of such new environment is defined.     

Idempotent splittings,colimit completion,and weak aspects of the theory of monads

We show that some recent constructions in the literature, named ‘weak’ generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.     

An extensive-form analysis for comparing fuzzy information systems by means of the worth and quietness of information

Statistical decision problems deal with the choice among several actions, whose consequences depend upon the state of nature, and this choice is usually made on the basis of the outcome of an experiment selected from a collection of experiments whose probability measures depend upon the state. The selection criteria among the experiments are generally based on some measures of the information contained in each experiment about that true state.In this paper we suggest a new selection criterion among the experiments associated with a statistical decision problem, when the experimental outcomes cannot be exactly perceived by the decision maker, but rather the available information from each experimental outcome can be regarded as an element in a fuzzy information system (as defined by Tanaka, Okuda and Asai).In a previous paper we have proposed a method for comparing fuzzy information systems on the basis of the maximization of the ‘oworth of information of a fuzzy park information system’ (Tanaka et al). When the application of such a method leads to indifference between two fuzzy information systems, this indifference can be avoided by applying the criterion in this paper.This selection criterion is an extensive-form analysis which is based on two measures of the information contained in a fuzzy information system about the true state: the ‘worth of information of a fuzzy information system’ and the ‘expected quietness of information of a fuzzy information system’.The suitability of the criterion stated above is corroborated by studying its main properties and contrasting this procedure with other ones making use of different measures of information.