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Fuzzy Connections and Relations in Complete Residuated Lattices
Fuzzy Connections and Relations in Complete Residuated Lattices
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Dec, 13(4): 345-351
Copyright © 2013, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : February 01, 2013
  • Accepted : September 18, 2013
  • Published : December 25, 2013
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Yong Chan Kim

Abstract
In this paper, we investigate the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L . We give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, dual residuated) connections induced by L -fuzzy relations.
Keywords
1. Introduction
A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [1 - 10] . Hájek [11] introduced a complete residuated lattice L that is an algebraic structure for many-valued logic. A context consists of ( U, V, R ), where U is a set of objects, V is a set of attributes, and R is a relation between U and V . Bĕlohlávek [1 - 3] developed a notion of fuzzy contexts using Galois connections with R L X×Y on L .
In this paper, we investigate properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in L and give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, and dual residuated) connections induced by L -fuzzy relations.
Definition 1.1. [11 , 12] An algebra (L, ∧, ∨, ⊙, →, 0, 1) is called a complete residuated lattice if it satisfies the following conditions:
(C1) L = ( L ,≤,∨,∧, 1, 0) is a complete lattice with the greatest element 1 and the least element 0;
(C2) ( L , ⊙, 1) is a commutative monoid;
(C3) x y z iff x y z for x , y, z L .
Remark 1.2. [11 , 12] (1) A completely distributive lattice L = ( L , ≤, ∨, ∧ = ⊙, → 1, 0) is a complete residuated lattice defined by
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In particular, the unit interval ([0, 1],∨,∧ = ⊙, → 0, 1) is a complete residuated lattice defined by
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(2) The unit interval with a left-continuous t-norm ⊙, ([0, 1], ∨, ∧, ⊙, →, 0, 1) , is a complete residuated lattice defined by
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In this paper, we assume that ( L , ∧, ∨, ⊙, →, 0, 1) is a complete residuated lattice with the law of double negation, i.e., α = α ** where α = α → 0.
Lemma 1.3. [12] For each x, y, z, xi, yi , ∈ L , we have the following properties.
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Definition 1.4. [4 , 7] Let X denote a set. A function eX : X × X L is called:
(E1) reflexive if eX ( x, x ) = 1 for all x X ,
(E2) transitive if eX ( x, y ) ⊙ eX ( y, z ) ≤ eX ( x, z ), for all x, y, z X , and
(E3) if eX ( x, y ) = eX ( y, x ) = 1, then x = y .
If e satisfies (E1) and (E2), ( X, eX ) is a fuzzy preorder set. If e satisfies (E1), (E2), and (E3), ( X, eX ) is a fuzzy partially order set (for simplicity, fuzzy poset).
Example 1.5. (1) We define a function
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as
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Then, ( LX, eLX ) is a fuzzy poset from Lemma 1.3 (10, 11).
(2) If ( X, eX ) is a fuzzy poset and we define a function
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is a fuzzy poset.
2. Fuzzy Connections and Relations in Complete Residuated Lattices
Definition 2.1. Let ( X, eX ) and ( Y, eY ) denote fuzzy posets and f : X Y and g : Y X denote maps.
(1) ( eX, f, g, eY ) is called a Galois connection if for all x X, y Y ,
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(2) ( eX, f, g, eY ) is called a dual Galois connection if for all x X , y Y ,
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(3) ( eX, f, g, eY ) is called a residuated connection if for all x X , y Y ,
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(4) ( eX, f, g, eY ) is called a dual residuated connection if for all x X , y Y ,
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(5) f is an isotone map if eY ( f(x 1 ), f(x 2 )) ≥ eX ( x 1 , x 2 ) for all x 1 , x 2 X .
(6) f is an antitone map if eY ( f(x 1 ), f(x 2 )) ≥ eX ( x 2 , x 1 ) for all x 1 , x 2 X .
(7) f is an embedding map eY ( f(x 1 ), f(x 2 )) ≥ eX ( x 1 , x 2 ) for all x 1 , x 2 X .
If X = Y and eX = eY , we simply denote ( eX, f, g ) for ( eX, f, g, eY ). ( X , ( eX, f, g )) is called a Galois (resp. residuated, dual Galois, and dual residuated) pair.
Remark 2.2. Let ( X, eX ) and ( Y, eY ) denote a fuzzy poset and f : X Y and g : Y X denote maps.
(1) ( eX, f, g, eY ) is a Galois (resp. dual Galois) connection iff ( eY, g, f, eX ) is a Galois (resp. dual Galois) connection.
(2) ( eX, f, g, eY ) is a Galois (resp. residuated) connection iff
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is a dual (resp. dual residuated) Galois connection.
(3) ( eX, f, g, eY ) is a residuated (resp. dual residuated) connection iff
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is a residuated (resp. dual residuated) connection.
(4) ( eX, f, g, eY ) is a Galois (resp. dual Galois) connection iff
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is a residuated (resp. dual residuated) connection.
(5) ( eX, f, g, eY ) is a residuated connection iff ( eY, g, f, eX ) is a dual residuated connection.
Theorem 2.3. Let ( X, eX ) and ( Y, eY ) denote a fuzzy poset and f : X Y and g : Y → X denote maps.
(1) ( eX, f, g, eY ) is a Galois connection if f, g are antitone maps and eY (y, f(g(y ))) = eX(x, g(f(x) )) = 1.
(2) ( eX, f, g, eY ) is a dual Galois connection if f, g are antitone maps and eY(f(g(y)),y) = eX(g(f(x)), x) = 1.
(3) ( eX, f, g, eY ) is a residuated connection if f, g are isotone maps and eY(f(g(y)),y) = eX(x,g(f(x))) = 1.
(4) ( eX, f, g, eY ) is a dual residuated connection if f, g are isotone maps and eY(y,f(g(y))) = eX(g(f(x)),x) = 1.
Proof. (1) Let ( f, g ) denote a Galois connection. Since
  • eY (y, f(x)) = eX(x, g(y)),
we have
  • 1 =eY(f(x), f(x)) = eX(x, g(f(x)))
and
  • eY(y, f(g(y))) = eX(g(y), g(y)) = 1.
Furthermore,
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Conversely,
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Similarly, eY(y, f(x)) ≤ eX(x, g(y)).
(2) Since eY(f(x), y) = eX(g(y), x) , we have
  • 1 =eY(f(x), f(x)) = eX(g(f(x)), x)
and
  • eY(f(g(y)), y) = eX(g(y), g(y))= 1.
Furthermore,
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Conversely,
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Similarly, eY(f(x), y) ≤ eX(g(y), x).
(3) Since eY(f(x), y) = eX(x, g(y)) , we have
  • 1 =eY(f(x), f(x)) = eX(x, g(f(x)))
and
  • eY(f(g(y)), y) = eX(g(y), g(y))= 1.
Furthermore,
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Conversely,
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Moreover,
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(4) It is similarly proved as (3).
Example 2.4. Let X = {a, b, c} denote a set and f : X X denote a function as f(a) = b, f(b) = a, f(c) = c . Define a binary operation ⊙ (called Łukasiewicz conjunction) on L = [0, 1] using
  • x⊙y= max{0,x+y– 1},
  • x→y= min{1 –x+y, 1}.
(1) Let ( X = { a, b, c }, e 1 ) denote a fuzzy poset as follows:
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Since e1(x, y) = e1(f(x), f(y)),
  • e1(x, f(f(x))) = e1(f(f(x)), x) = 1,
then, ( e1; f, f ) are both residuated and dual residuated connections. Since 0:7 = e 1 ( c, a ) ≰ e 1 ( f(a), f(c )) = e 1 ( b, c ) = 0.5, f is not an antitone map. Hence, ( e 1 , f, f ) are neither Galois nor dual Galois connections.
(2) Let ( X = { a, b, c }, e 2 ) denote a fuzzy poset as follows:
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Since e 2 ( x, y ) = e 2 ( f(y), f(x )),
  • e2(x, f(f(x))) =e2(f(f(x)), x) = 1,
then, ( e2, f, f ) are both Galois and dual Galois connections. Since 0.7 = e 2 ( c,a ) ≰ e 2 ( f(c), f(a )) = e 2 ( c, b ) = 0.5, f is not an isotone map. Hence, ( e 2 , f, f ) are neither residuated nor dual residuated connections.
Definition 2.5. Let R LX×Y denote a fuzzy relation. For each A LX and B LY , we define operations R –1 ( y, x ) = R(x, y ) and
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as follows:
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Theorem 2.6. Let R LX×Y denote a fuzzy relation. For each A LX and B LY ,
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Proof. (1) From Lemma 1.3 (13,14), we have
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(3) From Lemma 1.3 (7), we have
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From Lemma 1.3 (8), we have
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(4) From Lemma 1.3 (8), we have
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Other cases are similarly proved.
Theorem 2.7. Let R L×Y denote a fuzzy relation, ( LX , eLX ) and ( LY , eLY ) denote fuzzy posets. We have the following properties.
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Proof. (1) For each C LX , B LY ,
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(2) For each C LX , B LY ,
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(3) For each C LX , B LY ,
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(4) For each C LX , B LY ,
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For each C LX , B LY ,
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Other cases are similarly proved.
3. Conclusion
In this paper, we investigated the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L . In particular, we studied fuzzy Galois (dual Galois, residuated, and dual residuated) connections induced by L -fuzzy relations.
In the future, we will investigate the properties using fuzzy connections on algebraic structures and study the fuzzy concept lattices.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
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