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 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 30, 2013 PDF e-PUB PubReader PPT Export by style
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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  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 Lager Image
In particular, the unit interval ([0, 1],∨,∧ = ⊙, → 0, 1) is a complete residuated lattice defined by Lager Image
(2) The unit interval with a left-continuous t-norm ⊙, ([0, 1], ∨, ∧, ⊙, →, 0, 1) , is a complete residuated lattice defined by Lager Image
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.  For each x, y, z, xi, yi , ∈ L , we have the following properties. Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image
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 Lager Image
as Lager Image
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 Lager Image
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 , Lager Image
(2) ( eX, f, g, eY ) is called a dual Galois connection if for all x X , y Y , Lager Image
(3) ( eX, f, g, eY ) is called a residuated connection if for all x X , y Y , Lager Image
(4) ( eX, f, g, eY ) is called a dual residuated connection if for all x X , y Y , Lager Image
(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 Lager Image
is a dual (resp. dual residuated) Galois connection.
(3) ( eX, f, g, eY ) is a residuated (resp. dual residuated) connection iff Lager Image
is a residuated (resp. dual residuated) connection.
(4) ( eX, f, g, eY ) is a Galois (resp. dual Galois) connection iff Lager Image
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, Lager Image
Conversely, Lager Image
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, Lager Image
Conversely, Lager Image
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, Lager Image
Conversely, Lager Image
Moreover, Lager Image
(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: Lager Image
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: Lager Image
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 Lager Image
as follows: Lager Image
Theorem 2.6. Let R LX×Y denote a fuzzy relation. For each A LX and B LY , Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image
Proof. (1) From Lemma 1.3 (13,14), we have Lager Image Lager Image
(3) From Lemma 1.3 (7), we have Lager Image
From Lemma 1.3 (8), we have Lager Image
(4) From Lemma 1.3 (8), we have Lager Image Lager Image Lager Image Lager Image Lager Image Lager Image
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. Lager Image
Proof. (1) For each C LX , B LY , Lager Image
(2) For each C LX , B LY , Lager Image
(3) For each C LX , B LY , Lager Image
(4) For each C LX , B LY , Lager Image
For each C LX , B LY , Lager Image
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