Galois Connections Between a Fuzzy Preordered Structure and a General Fuzzy Structure

Cabrera, Inma P.; Cordero, Pablo; García-Pardo, Francisca; Ojeda‐Aciego, Manuel; Baets, Bernard De

doi:10.1109/tfuzz.2017.2718495

Cited by 24 publications

(5 citation statements)

References 40 publications

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“…The consideration of negations in this study naturally suggests to further analyze the properties of the functional degrees in terms of the opposition square, which has recently been studied in the fuzzy framework [17,18], and in the framework of positive idempotent semifields where negation is carried out by inversion, see [19]. On the other hand, the results of Theorems 16 and 18 on the existence of isotone Galois connections (also called adjunctions) suggest to further investigate the possibility of defining extension of the proposed f-degrees in more general contexts, taking into account the characterisation of existence of adjoint mappings in different fuzzy settings [20,21].…”

Section: Discussionmentioning

confidence: 99%

On Contradiction and Inclusion Using Functional Degrees

Madrid¹,

Ojeda‐Aciego²

2020

IJCIS

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The notion of inclusion is a cornerstone in set theory and therefore, its generalization in fuzzy set theory is of great interest. The degree of f-inclusion is one generalization of such a notion that differs from others existing in the literature because the degree of inclusion is considered as a mapping instead of a value in the unit interval. On the other hand, the degree of f-weakcontradiction was introduced to represent the contradiction between two fuzzy sets via a mapping and its definition has many similarities with the f-degree of inclusion. This suggests the existence of relations between both f-degrees. Specifically, following this line, we analyze the relationship between the f-degree of inclusion and the f-degree of contradiction via the complement of fuzzy sets and Galois connections.

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Section: Discussionmentioning

confidence: 99%

On Contradiction and Inclusion Using Functional Degrees

Madrid¹,

Ojeda‐Aciego²

2020

IJCIS

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“…A usual way to define fuzzy algebras is to consider as an underlying structure a pair which consists of a set and a tolerance or equivalence relation on it. Thus, an alternative definition of fuzzy poset that can be found in the literature [9] is given by a tuple (A, ≈, ρ) where ≈ is a fuzzy equivalence relation on A and ρ is a fuzzy order that is compatible with ≈. In [24], it is shown that both definitions of fuzzy poset are equivalent.…”

Section: Preliminariesmentioning

confidence: 99%

“…We can find distinct definitions of closure system depending on the ordered structure on which the fuzzy closure operator is defined. As a consequence, the notion of fuzzy closure system has been defined on L-ordered sets [14], on fuzzy preposets [8] and fuzzy preordered structures [9]. This paper is a continuation on the study of fuzzy closure systems done in [19], where the underlying structure was a Heyting algebra.…”

Section: Introductionmentioning

confidence: 99%

Closure Systems as a Fuzzy Extension of Meet-subsemilattices

Ojeda-Hernández¹,

Cabrera²,

Muñoz‐Velasco³

2021

Atlantis Studies in Uncertainty Modelling

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An extension of the notion of closure system is studied adapting the idea of meetsubsemilattice to a complete fuzzy lattice. Results relating closure operators and closure systems in the classical case are extended properly to this framework. This definition is proved to be equivalent to the most used definition given by Bělohlávek on the fuzzy powerset lattice. A definition of fuzzy closure system is presented and related to closure systems. Atlantis Studies in Uncertainty Modelling, volume 3Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP)

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“…Galois functions were generalized to the fuzzy case by Bělohlávek [14]. Cabrera et al investigate Galois connections in the framework of fuzzy-preordered structures using particular fuzzy equivalence relations with a residuated lattice as the membership-values structure [15][16][17].…”

Section: Historical Remarksmentioning

confidence: 99%

Kernels of Residuated Maps as Complete Congruences in Lattices

Šešelja¹,

Tepavčević²

2020

IJCIS

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In a context of lattice-valued functions (also called lattice-valued fuzzy sets), where the codomain is a complete lattice L, an equivalence relation defined on L by the equality of related cuts is investigated. It is known that this relation is a complete congruence on the join-semilattice reduct of L. In terms of residuated maps, necessary and sufficient conditions under which this equivalence is a complete congruence on L are given. In the same framework of residuated maps, some known representation theorems for lattices and also for lattice-valued fuzzy sets are formulated in a new way. As a particular application of the obtained results, a representation theorem of finite lattices by meet-irreducible elements is given.

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Galois Connections Between a Fuzzy Preordered Structure and a General Fuzzy Structure

Cited by 24 publications

References 40 publications

On Contradiction and Inclusion Using Functional Degrees

On Contradiction and Inclusion Using Functional Degrees

Closure Systems as a Fuzzy Extension of Meet-subsemilattices

Kernels of Residuated Maps as Complete Congruences in Lattices

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