Costas A. Drossos

Correspondence analysis with fuzzy data: The fuzzy eigenvalue problem

This paper constitutes a first step towards an extension of correspondence analysis with fuzzy data (FCA). At this stage, our main objective is to lay down the algebraic foundations for this fuzzy extension of the usual correspondence analysis. A two-step method is introduced to convert the fuzzy eigenvalue problem to an ordinary one. We consider a fuzzy matrix as the set of its cuts. Each such cut is an interval-valued matrix viewed as a line-segment in the matrix space. In this way, line-segments of cut-matrices are transformed into intervals of eigenvalues. Therefore, the two-step method is essentially a reduction of the fuzzy eigenvalue problem to an ordinary one. We illustrate the FCA-fuzzy eigenvalue problem with a simple numerical example. We hope upon the completion of this project in near future, to be able to supply the necessary tools for the end user of the correspondence analysis with fuzzy data.

Foundations of fuzzy sets: a nonstandard approach

Abstract The well known results on the inner limitations of classical mathematics (Lowenheim-Skolem Theorems, and Godels Theorems) and the introduction of a host of new set-like concepts, indicate the presence of something which is beyond the grasp of Cantorian set theory. In different ways each limitation expresses a basic uncertainty in Mathematics. Godels Theorems essentially express the following: Either we restrict ourselves to classical mathematical objects and their axiomatic description, and have some limitation on the amount of information that can be derived from these axioms systems or we consider nonclassical objects (fuzzy or variable objects, vague predicates, etc.) and lose the exactness associated with their description and possibly the law of excluded middle associated with their logic. On the other hand the Lowenheim-Skolem theorems express a limitation on the capability of axiom systems to describe models uniquely, and therefore a kind of relativity of models, according to the method of ‘observation’ of the mathematical reality. It is argued in this paper that the common element of all new set-like concepts introduced is variability and vagueness. We start the study of vagueness by indicating its aspects, first in the ‘intentional’ development of Nonstandard Analysis (NSA), and then passing to the case of B -fuzzy sets. The nonstandard approach to Fuzzy Sets is based on the idea that, if we introduce a ‘local observer’ into ZFC, who ‘sees’ the absolute ZFC framework with a local and non-Cantorian way, then we get a fuzzy deformation of ZFC, which constitues ‘A Theory of Fuzzy Sets’. The introduction of the observer is signified by the presence of the predicate ‘standard’, or of a probability space, etc. This is a first of a series of papers applying nonstandard methods to the study of vagueness and fuzzy sets.

Boolean fuzzy sets

Abstract The classical theory of fuzzy sets has been developed within the Cantorian framework, as a simple generalization of the indicator or characteristic function and thus as a generalization of the membership property only, paying no attention to the equality relation. The nonstandard approach to fuzzy sets on the contrary is based on a non-Cantorian framework, which essentially amounts to viewing the eternally constant Cantorian-Platonic universe through two observers: the absolute Cantorian and the local non-Cantorian [5]. The construction of such a theory is based on a generalization of both equality and membership relations and technically is based on the theory of Boolean powers and a Boolean generalization of Nonstandard Analysis based on them [5, 6, 8, 15]. For Nonstandard Analysis see [1, 13]. In this paper, which is a continuation of [5], we give some new results for the basic concepts of fuzzy sets from our point of view, along with interpretations which are more interesting to fuzzy theorists.

Generalized t-norm structures

Abstract This is a continuation of the work in (Drossos and Navara, EUFIT 96, Aachen, Germany, 1996). In this paper we investigate the possibility of defining a quantitative structure (non-idempotent) on the top of a usual qualitative structure. These generalized monoidal structures are very much related to the notion of t-norms and t-conorms defined on the closed interval [0,1]. We would then like to define an appropriate generalization of the notion of t-norm, as a generalized monoidal structure, which will reflect a minimum of properties of t-norms defined on [0,1] considered as absolutely basic.

Coupling an MV-algebra with a Boolean algebra

Abstract In this article we study Boolean powers of MV-algebras. We show that the property of semisimplicity is preserved by Boolean powers over separable complete Boolean algebras or by bounded Boolean powers over any Boolean algebra. In addition we give an explicit method of construction, based on Boolean powers, of an MV-algebra with given center. This last construction allows one to combine a qualitative (idempotent) structure like the Boolean algebra with a quantitative (non-idempotemt) structure like the MV-algebra.

A Many-Valued Generalization of the Ultrapower Construction

D. Scott in [1969] remarked: ‘The idea of constructing Boolean-valued models could have been (but was not) discovered as a generalization of the ultraproduct method used now so often to obtain nonstandard models for ordinary analysis. Roughly, we can say that ultraproducts use the standard Boolean algebras (the power-set Boolean algebras) to obtain models elementarily equivalent to the standard model, whereas the Boolean method allows the nonstandard complete algebras (such us the Lebesgue algebra of measurable sets modulo sets of measure zero or the Baire algebra of Borel sets modulo sets of the first category.) Thus the Boolean method leads to nonstandard nonstandard models that are not only not isomorphic to the standard model but are not even equivalent. Nevertheless, they do satisfy all the usual axioms and deserve to be called models of analysis.’

Nonstandard methods in many-valued logics

Abstract Since MV-algebras include, in an essential way, infinitisimals it was felt that a systematic study of relationships of nonstandard methods and MV-algebras was needed. In this paper we try to introduce some basic frameworks to study MV-algebras from a nonstandard point of view. Also one can see that for a complete study of MV-algebras many external objects need to be treated differently.

A note on representing and interpreting MV-algebras

We try to make a distinction between the idea of representing and that of interpreting a mathematical structure. We present a slight generalization of Di Nolas Representation Theorem as to incorporate this point of view. Furthermore, we examine some preservation and functorial aspects of the Boolean power construction.