Y. Kawahara and H. Furusawa, An Algebraic Formalization of Fuzzy Relations, RIFIS Technical Report TR-CS-98, Kyushu University (1995).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....left to the user as an additional proof obligation. Ruby has adopted the component oriented view with relations as binary predicates, which makes it non applicable for abstract relation algebras in general. Recent research has attempted to combine fuzzy set theory with abstract relation algebra [14, 20]. Once axiomatized, the obtained fuzzy relation algebra can be used to represent informedness in data base semantics and is recommended to have a corresponding theorem proving system present. It seems to be a promising future research to extend the flexible RALL system in order to deal with fuzzy ....

Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Draft paper (April 19, 1995)

....algebras, have been investigated. Schmidt and Strohlein [6, 7] gave a simple proof of the representation theorem for Boolean relation algebras satisfying Tarski rule and a point axiom. A representation theorem for fuzzy relation algebras satisfying a point axiom was proved by Kawahara and Furusawa [1], and categorical representation theorems of fuzzy relations were proved by Kawahara, Furusawa and Mori [2] The investigation on fuzzy theory has begun by Zadeh in 1965. Then fuzzy relations have played an important role in mathematics, science and engineering. A methodology for processing fuzzy ....

....the greatest relation r. A representable relation algebra has no ideal relation except for the zero relation O and the greatest relation r. The aim of this paper is to provide an algebraic characterization of cartesian products of fuzzy relations by adding two axioms to a set of axioms given by [1]. One of axioms is called an axiom of cartesian products, introduced by J onsson and Tarski [9] and the other is called a point axiom for cartesian products. Ideal relations in fuzzy relation algebras are required to be crisp. The second one is defined by improving notion of point relations and ....

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Y. Kawahara and H. Furusawa, An Algebraic Formalization of Fuzzy Relations, RIFIS Technical Report TR-CS-98, Kyushu University (1995).

....style, which makes relational calculus a very useful framework for the study of mathematics [7, 11] and theoretical computer science [8, 13] and also a useful tool for applications. Some element free formalisations of fuzzy relations and proofs of representation theorems were provided in [4, 5, 6]. 1 2 Hitoshi Furusawa In this paper we consider relation algebras, which may not be Boolean, and provide their representation theorem. Relation algebras in the sense of this paper are equivalent to Dedekind categories (or allegories) with just one object. Kawahara, Furusawa, and Mori [5] proved ....

....included in the identity relation and which commutes with the greatest relation with respect to composition. In the case of L relations, scalar relations can be represented as scalar matrices. We use the concept of scalar relations to define a new concept of crisp relations different from that in [4, 5, 6]. Also the set of all scalar relations is a complete distributive lattice, which is a sublattice of the relation algebra, and scalar relations represent membership values. The concept of point relations was introduced by Schmidt and Strohlein in [8, 9] in the context of applications of (Boolean) ....

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Y. Kawahara and H. Furusawa, An Algebraic Formalization of Fuzzy Relations, to appear in Fuzzy Sets and Systems, 1997.

....all natural numbers n. Hence r = u n0 k n r = n0 k n )r = 0r = O by F2(d) and F2(a) e) This statement is obviously equivalent to (d) f) Assume kr v k 0 r and k 0 k. Then 0 k 0 =k 1 and kr v k[ k 0 =k)r] Therefore r v (k 0 =k)r by F2(e) and so r = O by (d) 2 Following to [5] the concept of crisp elements in fuzzy algebras is defined as follows: Definition 3.3 An element a of a fuzzy algebra F is crisp if a u kr = ka for all scalars k 2 [0; 1] 2 In the above definition of crisp elements of fuzzy algebras it is trivial that ka v a u kr by 3.2(a) and 3.2(b) Note ....

....X = ff u krXY ) kff) kff ] by 2.2(a) 2.2(b) and Z2(b) fffi u krXZ v ff[fi u ff ] krXZ ) v ff(fi u kr Y Z ) ff(kfi) k(fffi) by D3, 3.2(a) Z2(c) and (b) d) id X u krXX = id X u k(id X rXX ) id X u (kid X )rXX v (kid X ) kid X ) id X u rXX ] kid X . 2 In view of [5, 13] the concept of points in Zadeh categories is defined as follows: Definition 4.3 A point x of an object X is a crisp morphism x : X X such that x ] x v id X , id X v xx ] and rXX x = x. 2 Note that a point x of X is automatically nonzero from the totality id X v xx ] if X is nonempty. An ....

Y. Kawahara and H. Furusawa, An algebraic formalization of fuzzy relations, RIFIS-TRCS 98, Kyushu University, February 1995.

....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 free formalizations of fuzzy relations and proofs of representation theorems were provided in [3, 9, 10]. In this paper we consider Dedekind categories named by Olivier and Serrato [14] One of the aim of this paper is to study notions of crispness and scalar relations in Dedekind categories. A notion of crispness was introduced in [10] under the assumption that Dedekind categories have unit objects ....

....and Strohlein [18] and a point axiom will be stated to show a representation theorem in uniform Dedekind categories. In particular, the point axiom induces a function assigning a concrete L relation between the sets of point relations to an abstract relation in Dedekind categories. In view of [4, 9, 18] the concept of points in Dedekind categories is defined as follows: Definition 5.1 Let D be a Dedekind category. A point x of X is an s crisp function x : X X such that rXX x = x. 2 We will denote the set of all points of X by (X) Lemma 5.2 Let x and x 0 be points of X. Then (a) If rXX ae ....

Y. Kawahara and H. Furusawa, An algebraic formalization of fuzzy relations, To appear in Fuzzy Sets and Systems, 1997.

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