Categories of fuzzy sets

Walker, Carol L.

doi:10.1007/s00500-003-0275-1

Cited by 15 publications

(11 citation statements)

References 14 publications

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“…We introduce the concept of L-valued closure space, and L-closure of L-valued subgroup of a group in Section 4; we also introduce here a category of L-valued closure groups -a topological category. With the help of connections, as presented by L. N. Stout in [32] and C. L. Waker in [33] between the categories of L-SET and L-TOL, the category of L-valued tolerance spaces [32], we prove a connection between L-GRP, category of L-valued subgroups, and L-valued transitive tolerance spaces, L-TranTOL. Section 5 is devoted to study properties of L-valued closure of L-valued subgroups in the context of L-valued topological groups, where some properties from groups are taken into consideration.…”

Section: Introductionmentioning

confidence: 83%

“…[11,23,25,26,29] but its categorical behaviors are explored in a certain extent in recent times [26], although the category of fuzzy sets being studied for quite a long time, cf. [14,33]. In [3], we also considered L-valued closure of an L-valued subgroup of a group in the context of L-valued neighborhood groups, where the lattice under consideration was a complete MV-algebra with square roots.…”

Section: Introductionmentioning

confidence: 99%

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On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

Ahsanullah

Al-Thukair

2021

Eur. J. Pure Appl. Math.

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Starting with L as an enriched cl-premonoid, in this paper, we explore some categorical connections between L-valued topological groups and Kent convergence groups, where it is shown that every L-valued topological group determines a well-known Kent convergence group, and conversely, every Kent convergence group induces an L-valued topological group. Considering an L-valued subgroup of a group, we show that the category of L-valued groups, L-GRP has initial structure. Furthermore, we consider a category L-CLS of L-valued closure spaces, obtaining its relation with L-valued Moore closure, and provide examples in relation to L-valued subgroups that produce Moore collection. Here we look at a category of L-valued closure groups, L-CLGRP proving that it is a topological category. Finally, we obtain a relationship between L-GRP and L-TransTOLGRP, the category of L-transitive tolerance groups besides adding some properties of L-valued closures of L-valued subgroups on L-valued topological groups.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

Ahsanullah

Al-Thukair

2021

Eur. J. Pure Appl. Math.

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“…See, for example, [2], [5], [7], [8], [9]. We extend this to categories of interval-valued fuzzy sets with functions or relations, and lift elements of the structure on the truth-value algebra to the categorical setting.…”

Section: Categories Of Fuzzy Setsmentioning

confidence: 99%

“…Properties of Set ([0, 1]) shown in [9] generalize without difficulty to the category FIV. We give some informal definitions of some basic properties.…”

Section: Properties Of Fivmentioning

confidence: 99%

Categories of interval-valued fuzzy sets

Harding

Walker

2012

2012 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS)

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Abstract-We study two categories, both having intervalvalued fuzzy sets as objects. One has certain functions between the domains as morphisms and the other is expanded to include certain relations between the domains as morphisms. We describe some of the basic properties of each of these categories. We lift tnorms and negations to the category with relations. The t-norms are used to define a "tensor product" on that category. I. CATEGORIES OF FUZZY SETSA category consists of objects, morphisms, a composition rule for morphisms, and an identity morphism for each object. Some examples of categories are vector spaces with linear transformations; sets with functions; topological spaces with continuous functions; and groups with homomorphisms. In all these examples, composition is ordinary composition of functions. Another example, which we shall refer to, is the less familiar category having sets as objects and relations as morphisms, and using ordinary composition of relations.The study of categories of fuzzy sets with the truth-value algebra I = ([0, 1] , ∧, ∨, ¬, 1, 0), where the operations ∧ and ∨ are the usual min and max, negation ¬x = 1 − x, and constants 1 and 0, has a long history. See, for example, [2], [5], [7], [8], [9]. We extend this to categories of interval-valued fuzzy sets with functions or relations, and lift elements of the structure on the truth-value algebra to the categorical setting.The categories of fuzzy sets that we consider have as objects interval-valued fuzzy sets A, with is the truth-value algebra for interval-valued fuzzy sets. Some of the basic properties of this algebra may be found in [4]. Since the domains X, Y, ... of the fuzzy sets need to be identified in order to describe the morphisms between objects, and the range I[2] of the fuzzy sets will be always the same, we will write (X, A) or (Y, B) for objects X. Morphisms for these categories will be functions or relations satisfying the conditions described in 1) and 2), and composition is in both cases ordinary composition of functions or relations.[2] is a distributive lattice, in fact, a De Morgan algebra since the negation is an involution satisfying the De Morgan laws [1].We will consider two categories, FIV and FIVR, with objects interval-valued fuzzy sets and morphisms functions and relations, respectively:1) The objects of FIV are functions (X, A) = X A → I[2] , and the morphismsComposition is ordinary composition of functions and the identity id X :2) The objects of FIVR are functions (X, A) = X A → I [2] , and the morphismsComposition is ordinary composition of relations and the identity id X : X → X is the relation id X = {(x, x) : x ∈ X}. The inequality conditions in 1) and 2) will be indicated by diagrams of the form(where R could be a function or relation). For purposes of comparison, we will also consider the categories Set (the objects are sets and the morphisms are functions) and Rel (the objects are sets and the morphisms are relations R ⊆ X × Y ). We show that morphisms are closed under composition in FIVR.Lem...

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“…Several authors [2,6,7,21,23] have considered fuzzy sets as a category, which we will call FSet, where a morphism from A : X → I to B : Y → I is a function f : X → Y that satisfies A(x) ≤ (B •f )(x) for each x ∈ X. Here we continue this path of investigation.…”

Section: Introductionmentioning

confidence: 99%

Categories with fuzzy sets and relations

Harding¹,

Walker²,

Walker³

2014

Fuzzy Sets and Systems

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We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets.

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Categories of fuzzy sets

Cited by 15 publications

References 14 publications

On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

Categories of interval-valued fuzzy sets

Categories with fuzzy sets and relations

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