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Categorical Representation Theorems of Fuzzy Relations
 Proceedings of 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery (RSFD 96) 190197
, 1996
"... This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with supmin composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to onepoint set, and for Zadeh categories without unit ob ..."
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This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with supmin composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to onepoint set, and for Zadeh categories without unit objects. Keywords: fuzzy relations, relation algebras, representation theorem, Dedekind categories, Zadeh category. 1 Introduction Since Zadeh's invention the concept of fuzzy sets has been extensively investigated in mathematics, science and engineering. The notion of fuzzy relations is also a basic one in processing fuzzy information in relational structures, see e.g. Pedrycz [10]. Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [12]. On the other hand theory of relations, namely relational calculus, has a long history, see [8, 13, 14] for...
An Algebraic Characterization of Cartesian Products of Fuzzy Relations
 Bulletin of Informatics and Cybernetics 29
, 1996
"... This paper provides an algebraic characterization of mathematical structures formed by cartesian products of fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom was given by G. Schmidt and T. Str ..."
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This paper provides an algebraic characterization of mathematical structures formed by cartesian products of fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom was given by G. Schmidt and T. Strohlein, and cartesian products of Boolean relation algebras were investigated by B. J'onsson and A. Tarski. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semiscalar multiplication. First we present a set of axioms for fuzzy relation algebras and add axioms for cartesian products of fuzzy relation algebras. Second we improve the definition of point relations. Then a representation theorem for such relation algebras is deduced. Keywords : fuzzy relations, cartesian products, relation algebras, representation theorem. 1 Introduction In 1941 Tarski [8] proposed a problem, that is, "Is every relation algebra isomorphic to an algebra of all Bo...
A Representation Theorem for Relation Algebras: Concepts of Scalar Relations and Point Relations
 Bulletin of Informatics and Cybernetics 30
, 1997
"... This paper provides a proof of a representation theorem for homogeneous relation algebras by using concepts of scalar relations and point relations. Keywords : Lrelations, relation algebras, scalar relations, point relations, representation theorem. 1 Introduction Just after Zadeh's work on fuz ..."
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This paper provides a proof of a representation theorem for homogeneous relation algebras by using concepts of scalar relations and point relations. Keywords : Lrelations, relation algebras, scalar relations, point relations, representation theorem. 1 Introduction Just after Zadeh's work on fuzzy sets in 1965, Goguen [1] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices, and also stressed the importance of relations as follows: The importance of relations is almost selfevident. Science is, in a sense, the discovery of relations between observables. Zadeh has shown the study of relations to be equivalent to the general study of systems (a system is a relation between an input space and an output space). The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski; see [12] for details of the history of the study of Boolean relation algebras. In [10] Tarski proposed a formalisation of Boolean relation ...
Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems
, 1998
"... The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representati ..."
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The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...
Goguen Categories
"... In a wide variety of problems one has to treat uncertain or incomplete information. Some kind of exact science is needed to describe and understand existing methods and to develop new attempts. Especially in applications of computer science, this is a fundamental problem. To handle such kind of info ..."
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In a wide variety of problems one has to treat uncertain or incomplete information. Some kind of exact science is needed to describe and understand existing methods and to develop new attempts. Especially in applications of computer science, this is a fundamental problem. To handle such kind of information, Zadeh [23] introduced the concept of fuzzy sets and relations. In contrast to usual sets, fuzzy sets are characterized by a membership relation taking its values from the unit interval [0, 1] of the real numbers. After its introduction in 1965 the theory of fuzzy sets and relations was ranked to be some exotic field of research. The success during the last years with even consumer products involving fuzzy methods causes a rapidly growing interest of engineers and computer scientists in this field. Nevertheless, Goguen [5] generalized this concept in 1967 to Lfuzzy sets and relations for an arbitrary complete Brouwerian lattice L instead of the unit interval [0, 1] of the real numbers. He described one of his motivating examples as follows: A housewife faces a fairly typical optimization problem in her grocery shopping: she must select among all possible grocery bundles one that meets as well as several criteria of optimality, such as cost, nutritional value, quality, and variety. The partial ordering of the bundles is an intrinsic quality of this problem. (Goguen [5], 1967)