Reinterpreting a fuzzy subset by means of a Sincov's functional equation

Campión, María Jesús; Catalán, Raquel García; Induráin, Esteban; Ochoa, Gustavo

doi:10.3233/ifs-131005

Cited by 5 publications

(6 citation statements)

References 24 publications

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“…Thus µX 1 and µX 2 coincide, so that X1 and X2 are indeed the same fuzzy set of the universe U . (See also Corollary 3.6 in [6]).…”

Section: Corollary Let U Be a Universe Letmentioning

confidence: 88%

“…Thus, the concept of a fuzzy set can actually be regarded from various alternative points of view, as functional equations (see [6,5]), nested families of sets and nested topologies (see [1,7,20]) or representable total preorders (see [12,6]) among others. In addition, in [8] the concept of a weightable quasi-metric was shown to be closely related to some classical functional equations, of which some were also intimately linked to the denition of a fuzzy set of a universe as proved in [6,5]. However, to complete the panorama, a direct link between the denition of a fuzzy set of a universe and the possibility of endowing that universe with a suitable weightable quasi-metric had not been explored yet.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the disymmetry function of a weightable quasi-metric satises Sincov's functional equation and induces a representable total preorder. But Sincov's functional equation has been also proved in [6] to be closely related to the classical denition of a fuzzy set of a universe. We may then conclude, that a weightable quasi-metric could be then directly retrieved from the mere denition of a fuzzy set, by means of an associated real-valued bivariate function that satises Sincov's functional equation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weightable quasi-metrics related to fuzzy sets

Campión¹,

Catalán²,

Induráin³

et al. 2017

HJMS

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“…Thus µX 1 and µX 2 coincide, so that X1 and X2 are indeed the same fuzzy set of the universe U . (See also Corollary 3.6 in [6]).…”

Section: Corollary Let U Be a Universe Letmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weightable quasi-metrics related to fuzzy sets

Campión¹,

Catalán²,

Induráin³

et al. 2017

HJMS

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“…Thus, in particular 0 ∈ F(A, A) for all A ⊆ X. For some more recent investigations, see [4,33,38,27,34,35,7,8,3,14,13]. The most relevant ones are the set-valued considerations of Smajdor [38] and Augustová and Klapka [3].…”

Section: Theoremmentioning

confidence: 99%

Composition iterates, Cauchy, translation, and Sincov inclusions

Fechner

Száz

2020

Acta Universitatis Sapientiae, Mathematica

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Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕn of a relation ϕ on X, defined by {\varphi ^0} = {\Delta _x},\,\,{\varphi ^n} = \varphi \circ {\varphi ^{n - 1}}{\rm{ if n}} \in \mathbb{N,}\,\,{\rm{and }}\,\,{\varphi ^\infty } = \bigcup\limits_{n = 0}^\infty {{\varphi ^n}} .In particular, by using the relational inclusion ϕn◦ϕm⊆ ϕn+m with n, m ∈ \mathbb{\bar {N}_0}}, we show that the function α, defined by \alpha \left( n \right) = {\varphi ^{\rm{n}}}\,\,\,{\rm{for n}} \in {{\rm\mathbb{\bar N}}_{\rm{0}}}, satisfies the Cauchy problem \alpha \left( n \right) \circ \alpha \left( {\rm{m}} \right) \subseteq \alpha \left( {{\rm{n}} + {\rm{m}}} \right),\,\,\,\alpha \left( 0 \right) = {\Delta _{\rm{x}}}.Moreover, the function f, defined by {\rm{f}}\left( {{\rm{n}},{\rm{A}}} \right) = \alpha \left( {\rm{n}} \right)\left[ {\rm{A}} \right]\,\,\,{\rm{for}}\,{\rm{n}} \in {{\rm\mathbb{\bar {N}}}_{\rm{0}}}\,\,{\rm{and}}\,{\rm{A}} \subseteq {\rm{X,}} satisfies the translation problem {\rm{f}}\left( {{\rm{n}},f(m,{\rm{A)}}} \right) \subseteq {\rm{f}}\left( {{\rm{n}} + {\rm{m,A}}} \right),\,\,\,{\rm{f}}\left( {0,{\rm{A}}} \right) = {\rm{A}}{\rm{.}}Furthermore, the function F, defined by {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) = \left\{ {{\rm{n}} \in {{{\rm\mathbb{\bar {N}}}}_{\rm{0}}}:\,\,{\rm{A}} \subseteq {\rm{f}}\left( {{\rm{n}},{\rm{B}}} \right)} \right\}\,\,{\rm{for}}\,\,{\rm{A,B}} \subseteq {\rm{X,}} satisfies the Sincov problem {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) + {\rm{F}}\left( {{\rm{B}},{\rm{C}}} \right) \subseteq {\rm{F}}\left( {{\rm{A,C}}} \right),\,\,\,\,0 \in {\rm{F}}\left( {{\rm{A}},{\rm{A}}} \right).Motivated by the above observations, we investigate a function F on the product set X2 to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{F}}\left( {{\rm{y}},{\rm{z}}} \right) \subseteq {\rm{F}}\left( {{\rm{x}},{\rm{z}}} \right) for all x, y, z ∈ X. For this, we introduce the convenient notations {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{y}},{\rm{x}}} \right)\,\,\,{\rm{and}}\,\,{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right), and \Phi \left( {\rm{x}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{x}}} \right)\,\,{\rm{and}}\,\,\Psi \left( {\rm{x}} \right)\bigcup\limits_{{\rm{y}} \in {\rm{X}}} {{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right).}Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric, or single-valued.

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“…Let F : X × X → R be a real-valued bivariate function defined on X. The function F satisfies the Sincov functional equation if F (x, y) + F (y, z) = F (x, z) holds for every x, y, z ∈ X (see [4,10]).…”

Section: Definition 2 a Preordermentioning

confidence: 99%

On the Structure of Acyclic Binary Relations

Alcantud

Campión

Candeal

et al. 2018

Communications in Computer and Information Science

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We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.

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Reinterpreting a fuzzy subset by means of a Sincov's functional equation

Cited by 5 publications

References 24 publications

Weightable quasi-metrics related to fuzzy sets

Weightable quasi-metrics related to fuzzy sets

Composition iterates, Cauchy, translation, and Sincov inclusions

On the Structure of Acyclic Binary Relations

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