Fuzzy Connections and Relations in Complete Residuated Lattices
 Author: Kim Yong Chan
 Organization: Kim Yong Chan
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue4, p345~351, 25 Dec 2013

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 byL fuzzy relations.

KEYWORD
Fuzzy Galois connections , Fuzzy posets , Fuzzy isotone maps , Fuzzy antitone maps

1. Introduction
A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [110]. Hájek [11] introduced a complete residuated lattice
L that is an algebraic structure for manyvalued logic. A context consists of (U, V, R ), whereU is a set of objects,V is a set of attributes, andR is a relation betweenU andV . Bĕlohlávek [13] developed a notion of fuzzy contexts using Galois connections withR ∈L ^{X×Y} onL .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 byL fuzzy relations.Definition 1.1. [11, 12] An algebra is called a complete residuated lattice if it satisfies the following conditions:(C1) 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 is a complete residuated lattice defined byIn particular, the unit interval is a complete residuated lattice defined by
(2) The unit interval with a leftcontinuous tnorm , , is a complete residuated lattice defined by
In this paper, we assume that is a complete residuated lattice with the law of double negation, i.e.,
a =a ** wherea =a → 0.Lemma 1.3. [12] For eachx, y, z, x_{i}, y_{i} ∈L , we have the following properties.Definition 1.4. [4, 7] LetX denote a set. A functione_{X} :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, e_{X} ) is a fuzzy preorder set. Ife satisfies (E1), (E2), and (E3), (X, e_{X} ) is a fuzzy partially order set (for simplicity, fuzzy poset).Example 1.5. (1) We define a functione_{L}x :L^{X} ×L^{X} →L as
e_{L}x (A, B ) = (A (x ) → B(x )).Then, (
L^{X}, e_{L} x) is a fuzzy poset from Lemma 1.3 (10, 11).(2) If (
X, e_{X} ) is a fuzzy poset and we define a function (x, y ) =e_{X} (y, x ), then (X , ) is a fuzzy poset.2. Fuzzy Connections and Relations in Complete Residuated Lattices
Definition 2.1. Let (X, e_{X} ) and (Y, e_{Y} ) denote fuzzy posets andf :X →Y andg :Y →X denote maps.(1) (
e_{X}, f, g, e_{Y} ) is called a Galois connection if for allx ∈X ,y ∈Y ,eY(y, f(x)) = eX(x, g(y)).
(2) (
e_{X}, f, g, e_{Y} ) is called a dual Galois connection if for allx ∈X ,y ∈Y ,eY(f(x), y) = eX(g(y), x).
(3) (
e_{X}, f, g, e_{Y} ) is called a residuated connection if for allx ∈X ,y ∈Y ,eY(f(x), y) = eX(x, g(y)).
(4) (
e_{X}, f, g, e_{Y} ) is called a dual residuated connection if for allx ∈X ,y ∈Y ,eY(y, f(x)) = eX(g(y), x).
(5)
f is an isotone map ife_{Y} (f (x _{1}),f (x _{2})) ≥e_{X} (x _{1},x _{2}) for allx _{1},x _{2} ∈X .(6)
f is an antitone map ife_{Y} (f (x _{1}),f (x _{2})) ≥e_{X} (x _{2},x _{1}) for allx _{1},x _{2} ∈X .(7)
f is an embedding map ife_{Y} (f (x _{1}),f (x _{2})) =e_{X} (x _{1},x _{2}) for allx _{1},x _{2} ∈X .If
X =Y ande_{X} =e_{Y} , we simply denote (e_{X}, f, g ) for (e_{X}, f, g, e_{Y} ). (X , (e_{X}, f, g )) is called a Galois (resp. residuated, dual Galois, and dual residuated) pair.Remark 2.2. Let (X, e_{X} ) and (Y, e_{Y} ) denote a fuzzy poset andf :X →Y andg :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 (, f, g, ) is a dual (resp. dual residuated) Galois connection. (3) (eX, f, g, eY ) is a residuated (resp. dual residuated) connection iff (, g, f, ) is a residuated (resp. dual residuated) connection. (4) (eX, f, g, eY) is a Galois (resp. dual Galois) connection iff (eX, f, g, ) 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, e_{X} ) and (Y, e_{Y} ) denote a fuzzy poset andf :X →Y andg :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. Sincee_{Y} (y ,f (x )) =e_{X} (x ,g (y )),we have
1 =
e_{Y} (f (x ),f (x )) =e_{X} (x ,g (f (x )))and
e_{Y} (y ,f (g (y ))) =e_{X} (g (y ),g (y )) = 1.Furthermore,
Conversely,
Similarly,
e_{Y} (y ,f (x )) ≤e_{X} (x ,g (y )).(2) Since
e_{Y} (f (x ),y ) =e_{X} (g (y ),x ), we have1 =
e_{Y} (f (x ),f (x )) =e_{X} (g (f (x )),x )and
e_{Y} (f (g (y )),y ) =e_{X} (g (y ),g (y )) = 1.Furthermore,
Conversely,
Similarly,
e_{Y} (f (x ),y ) ≤e_{X} (g (y ),x ).(3) Since
e_{Y} (f (x ),y ) =e_{X} (x ,g (y )), we have1 =
e_{Y} (f (x ),f (x )) =e_{X} (x ,g (f (x )))and
e_{Y} (f (g (y )),y ) =e_{X} (g (y ),g (y )) = 1.Furthermore,
Conversely,
Moreover,
(4) It is similarly proved as (3).
Example 2.4. LetX = {a, b, c } denote a set andf :X →X denote a function asf (a ) =b ,f (b ) =a ,f (c ) =c . Define a binary operation (called Łukasiewicz conjunction) onL = [0, 1] usingx 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:Since
e _{1}(x, y ) =e _{1}(f (x ),f (y )),e _{1}(x ,f (f (x ))) =e _{1}(f (f (x )),x ) = 1,then, (
e _{1},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:Since
e _{2}(x, y ) =e _{2}(f (y ),f (x )),e _{2}(x ,f (f (x ))) =e _{2}(f (f (x )),x ) = 1,then, (
e _{2},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. LetR ∈L ^{X×Y} denote a fuzzy relation. For eachA ∈L^{X} andB ∈L^{Y} , we define operationsR ^{−1}(y, x ) =R (x, y ) and :L^{Y} →L^{X} as follows:Theorem 2.6. LetR ∈L ^{X×Y} denote a fuzzy relation. For eachA ∈L^{X} andB ∈L^{Y} ,Proof. (1) From Lemma 1.3 (13,14), we have(2)
(3) From Lemma 1.3 (7), we have
From Lemma 1.3 (8), we have
(4) From Lemma 1.3 (8), we have
(6)
(7)
(9) from:
(10) from:
Similarly, from:
Other cases are similarly proved.
Theorem 2.7. LetR ∈L ^{X×Y} denote a fuzzy relation, (L^{X} ,e_{L} x) and (L^{Y} ,e_{L}ʏ ) denote fuzzy posets. We have the following properties.(1) and are residuated connections.
(2) and are dual residuated connections.
(3) and are Galois connections.
(4) and are dual Galois connections.
Proof. (1) For eachC ∈L^{X} ,B ∈L^{Y} ,(2) For each
C ∈L^{X} ,B ∈L^{Y} ,(3) For each
C ∈L^{X} ,B ∈L^{Y} ,(4) For each
C ∈L^{X} ,B ∈L^{Y} ,For each
C ∈L^{X} ,B ∈L^{Y} ,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 byL 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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