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11
Container Types Categorically
, 2000
"... A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a ..."
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Cited by 12 (0 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...
An Algebraic Formalization of Fuzzy Relations
 Fuzzy Sets and Systems 101
, 1995
"... This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by G. Schmidt and T. Strohlein. Unlike Boolean ..."
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Cited by 9 (5 self)
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This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by G. Schmidt and T. Strohlein. 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 improve the definition of point relations. Then by using relational calculus a representation theorem for such relation algebras is deduced without Tarski rule. Keywords : fuzzy relations, relation algebras, relational calculus, representation theorem. 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 [9]. Gogue...
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 7 (7 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Relational Set Theory
 Lecture Notes in Computer Science
, 1995
"... This article presents a relational formalization of axiomatic set theory, including socalled ZFC and the antifoundation axiom (AFA) due to P. Aczel. The relational framework of set theory provides a general methodology for the fundamental study on computer and information sciences such as theory of ..."
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Cited by 6 (4 self)
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This article presents a relational formalization of axiomatic set theory, including socalled ZFC and the antifoundation axiom (AFA) due to P. Aczel. The relational framework of set theory provides a general methodology for the fundamental study on computer and information sciences such as theory of graph transformation, situation semantics and analysis of knowledge dynamics in distributed systems. To demonstrate the feasibility of relational set theory some fundamental theorems of set theory, for example, CantorBernstein Schroder theorem, Cantor's theorem, Rieger's theorem and Mostowski's collapsing lemma are proved. 1 Introduction The study on (binary) relations on sets has been begun together with the pioneering works of set theory and since then theory of relations has been extensively investigated by many mathematicians from the view points of logic, algebra, topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux [14] and...
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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Cited by 6 (6 self)
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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...
Universal properties of Span
 in The Carboni Festschrift, Theory and Applications of Categories 13 (2005
"... Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphi ..."
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Cited by 4 (2 self)
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Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.
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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Cited by 3 (2 self)
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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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Cited by 2 (1 self)
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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...
SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES Dedicated to Dominique Bourn on the occasion of his 60th birthday
"... Abstract. The purpose of this paper is to prove a new, incompleterelative, version of Nonabelian Snake Lemma, where “relative ” refers to a distinguished class of normal epimorphisms in the ground category, and “incomplete ” refers to omitting all completeness/cocompleteness assumptions not involv ..."
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Cited by 1 (0 self)
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Abstract. The purpose of this paper is to prove a new, incompleterelative, version of Nonabelian Snake Lemma, where “relative ” refers to a distinguished class of normal epimorphisms in the ground category, and “incomplete ” refers to omitting all completeness/cocompleteness assumptions not involving that class. 1.
Quantum Logic in Dagger Categories with Kernels
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
Abstract
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1