## Categorical Representation Theorems of Fuzzy Relations (1996)

Venue: | Proceedings of 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery (RSFD 96) 190--197 |

Citations: | 6 - 6 self |

### BibTeX

@INPROCEEDINGS{Kawahara96categoricalrepresentation,

author = {Yasuo Kawahara and Hithoshi Furusawa and Masao Mori},

title = {Categorical Representation Theorems of Fuzzy Relations},

booktitle = {Proceedings of 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery (RSFD 96) 190--197},

year = {1996},

pages = {pages}

}

### OpenURL

### Abstract

This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to one-point 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...

### Citations

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On the calculus of relations
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(Show Context)
Citation Context ...ory of relations, namely relational calculus, has a long history, see [8, 13, 14] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski =-=[15]-=-. Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, which they called allegori... |

66 | Relations and Graphs – Discrete Mathematics for Computer Scientists - Schmidt, Ströhlein - 1993 |

59 |
Resolution of composite fuzzy relation equations
- Sanchez
- 1976
(Show Context)
Citation Context ...oguen [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 more details on the history. Almost modern formalizations of relation algebras are affected... |

40 | The origin of relation algebras in the development and axiomatization of the calculus of relations
- Maddux
- 1991
(Show Context)
Citation Context ... 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 more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15]. Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of add... |

15 |
Squares and rectangles in relation categories -- three cases: Semilattice, distributive lattice and boolean non-unitary. Fuzzy Sets and Systems
- Olivier, Serrato
- 1995
(Show Context)
Citation Context ...ones. Instead of Schroder categories we 3 Department of Informatics, Kyushu University 33, Fukuoka 812-81, Japan. 1 need to adopt Dedekind categories, named after Jean-Pierre Olivier and Dany Serrato =-=[9]-=-, in order to study fuzzy relation categories. This paper is organized as follows: In section 2 we first recall the fundamentals on relation category, which we will call Dedekind categories following ... |

15 |
Relation algebras --- concept of points and representability
- Schmidt, Strohlein
- 1985
(Show Context)
Citation Context ... 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 more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15]. Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of add... |

12 |
An algebra of additive relations
- Lane
- 1961
(Show Context)
Citation Context ...s, namely relational calculus, has a long history, see [8, 13, 14] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15]. Mac Lane =-=[7]-=- and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, which they called allegories. Concerning... |

11 |
Processing in relational structures: Fuzzy relational equations
- Pedrycz
- 1991
(Show Context)
Citation Context ...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 Sanch... |

10 |
Pushout-Complements and Basic Concepts of Grammars in Toposes
- Kawahara
- 1990
(Show Context)
Citation Context ...elation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [14] on relations and graphs with many useful examples in computer science. Also Kawahara and Mizoguchi =-=[3, 6, 4]-=- developed relational methodology for assertion semantics of programs, theory of graph transformations (or graph grammars) and categorical set theory. A representation theorem for (homogeneous) fuzzy ... |

9 | An algebraic formalization of fuzzy relations
- Kawahara, Furusawa
- 1999
(Show Context)
Citation Context ...a). (e) This statement is obviously equivalent to (d). (f) Assume kr v k 0 r and k 0 ! k. Then 0sk 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 o... |

9 |
Relational structures and their partial morphisms in the view of single pushout rewriting
- Kawahara, Mizoguchi
- 1994
(Show Context)
Citation Context ...elation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [14] on relations and graphs with many useful examples in computer science. Also Kawahara and Mizoguchi =-=[3, 6, 4]-=- developed relational methodology for assertion semantics of programs, theory of graph transformations (or graph grammars) and categorical set theory. A representation theorem for (homogeneous) fuzzy ... |

8 |
Korrespondenzen in abelschen
- Puppe
- 1962
(Show Context)
Citation Context ...tional calculus, has a long history, see [8, 13, 14] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15]. Mac Lane [7] and Puppe =-=[11]-=- exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, which they called allegories. Concerning with applicati... |

7 | Relational set theory
- Kawahara
- 1995
(Show Context)
Citation Context ...elation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [14] on relations and graphs with many useful examples in computer science. Also Kawahara and Mizoguchi =-=[3, 6, 4]-=- developed relational methodology for assertion semantics of programs, theory of graph transformations (or graph grammars) and categorical set theory. A representation theorem for (homogeneous) fuzzy ... |

6 |
Categories, allegories (North-Holland
- Freyd, Scedrov
- 1990
(Show Context)
Citation Context ...lmost modern formalizations of relation algebras are affected by the work of Tarski [15]. Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov =-=[1]-=- developed and summarized categorical relational calculus, which they called allegories. Concerning with applications to the relational theory of graphs and programs, Schmidt and Strohlein [13] gave a... |