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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...
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 ...
Crispness in Dedekind Categories
"... . This paper studies notions of scalar relations and crispness of relations. 1 Introduction Just after Zadeh's invention of the concept of fuzzy sets [19], Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relat ..."
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. This paper studies notions of scalar relations and crispness of relations. 1 Introduction Just after Zadeh's invention of the concept of fuzzy sets [19], Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relations, namely relational calculus, has been investigated since the middle of the nineteen century, see [13, 16, 17] for more details. Almost all modern formalisations of relation algebras are affected by the work of Tarski [18]. Mac Lane [12] and Puppe [15] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which they called allegories. In relational calculus one calculates with relations in an elementfree style, which makes relational calculus a very useful framework for the study of mathematics [8] and theoretical computer science [1, 7, 11] and also a useful tool for applications. Some element...
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...