Fuzzy Connections and Relations in Complete Residuated Lattices

Vol. 13, No. 4, December 2013, pp. 345-351 http://dx.doi.org/10.5391/IJFIS.2013.13.4.345

Fuzzy Connections and Relations in Complete Residuated Lattices

Yong Chan Kim

Department of Mathematics, Gangneung-Wonju National University, Gangneung, Korea

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.

Keywords: Fuzzy Galois connections, Fuzzy posets, Fuzzy isotone maps, Fuzzy antitone maps

Received: Feb. 1, 2013 Revised : Sep. 3, 2013 Accepted: Sep. 18, 2013 Correspondence to: Yong Chan Kim (yck@gwnu.ac.kr)

©The Korean Institute of Intelligent Systems

cc This is an Open Access article dis- tributed under the terms of the Creative Commons Attribution Non-Commercial Li- cense (http://creativecommons.org/licenses/

by-nc/3.0/) which permits unrestricted non- commercial use, distribution, and reproduc- tion in any medium, provided the original work is properly cited.

1. Introduction

A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [1-10]. H´ajek [11] introduced a complete residuated latticeLthat is an algebraic structure for many-valued logic. A context consists of(U, V, R), whereU is a set of objects,V is a set of attributes, andRis a relation betweenUandV. Bˇelohl´avek [1-3]

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 inLand 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(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 element1and the least element0;

(C2) (L,,1)is a commutative monoid;

(C3) xy≤ziffx≤y→zforx, y, z∈L.

Remark 1.2. [11, 12] (1) A completely distributive latticeL= (L,≤,∨,∧=,→,1,0)is a complete residuated lattice defined by

x→y=_

{z|x∧z≤y}.

In particular, the unit interval ([0,1],∨,∧ = ,→,0,1)is a complete residuated lattice defined by

x→y=_

{z|x∧z≤y}.

(2) The unit interval with a left-continuous t-norm,([0,1],∨,∧,,→,0,1), is a complete residuated lattice defined by

x→y=_

{z|xz≤y}.

In this paper, we assume that(L,∧,∨,,→,0,1)is a com- plete residuated lattice with the law of double negation, i.e., a=a^∗∗wherea=a→0.

Lemma 1.3. [12] For each x, y, z, xi, yi ∈ L, we have the following properties.

(1) Ify ≤ z,(xy) ≤ (xz),x → y ≤ x → z, and z→x≤y→x.

(2) xy≤x∧y≤x∨y.

(3) x→(V

i∈Γyi) =V

i∈Γ(x→yi)and(W

i∈Γxi)→y= V

i∈Γ(xi→y).

(4) x→(W

i∈Γyi)≥W

i∈Γ(x→yi).

(5) (V

i∈Γxi)→y≥W

i∈Γ(xi→y).

(6) (xy)→z=x→(y→z) =y→(x→z).

(7) x(x→y)≤y,x→y≤(y →z)→(x→z), and x→y≤(z→x)→(z→y).

(8) y≤x→(xy)andx≤(x→y)→y.

(9) x→y≤(xz)→(yz).

(10) (x→y)(y →z)≤x→z.

(11) x→y= 1iffx≤y.

(12) x→y=y^∗→x^∗.

(13) (xy)^∗=x→y^∗=y→x^∗andx→y= (xy^∗)^∗. (14) V

i∈Γx^∗_i = (W

i∈Γxi)^∗andW

i∈Γx^∗_i = (V

i∈Γxi)^∗. Definition 1.4. [4, 7] LetX denote a set. A functione_X : X×X→Lis called:

(E1) reflexive ifeX(x, x) = 1for allx∈X,

(E2) transitive if e_X(x, y)e_X(y, z) ≤ e_X(x, z), for all x, y, z∈X, and

(E3) ifeX(x, y) =eX(y, x) = 1, thenx=y.

Ifesatisfies (E1) and (E2),(X, e_X)is a fuzzy preorder set. Ife 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 e_LX :L^X×L^X →L as

e_LX(A, B) = ^

x∈X

(A(x)→B(x)).

Then,(L^X, e_LX)is a fuzzy poset from Lemma 1.3 (10, 11).

(2) If(X, e_X)is a fuzzy poset and we define a function e⁻¹_X (x, y) =eX(y, x), then(X, e⁻¹_X )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 andf :X→Y andg:Y →Xdenote maps.

(1) (e_X, f, g, e_Y)is called a Galois connection if for allx∈ X, y∈Y,

e_Y(y, f(x)) =e_X(x, g(y)).

(2) (e_X, f, g, e_Y)is called a dual Galois connection if for all x∈X, y∈Y,

e_Y(f(x), y) =e_X(g(y), x).

(3) (e_X, f, g, e_Y)is called a residuated connection if for all x∈X, y∈Y,

e_Y(f(x), y) =e_X(x, g(y)).

(4) (e_X, f, g, e_Y)is called a dual residuated connection if for allx∈X, y∈Y,

e_Y(y, f(x)) =e_X(g(y), x).

(5) f is an isotone map ifeY(f(x1), f(x2))≥eX(x1, x2) for allx1, x2∈X.

(6) f is an antitone map ife_Y(f(x₁), f(x₂))≥e_X(x₂, x₁) for allx1, x2∈X.

(7) fis an embedding map ifeY(f(x1), f(x2)) =eX(x1, x2) for allx₁, x₂∈X.

If X = Y ande_X = e_Y, we simply denote(e_X, f, g) for (eX, f, g, eY).(X,(eX, f, g))is called a Galois (resp. residu-

ated, dual Galois, and dual residuated) pair.

Remark 2.2. Let(X, eX)and(Y, eY)denote a fuzzy poset andf :X→Y andg:Y →Xdenote maps.

(1) (e_X, f, g, e_Y)is a Galois (resp. dual Galois) connection iff(eY, g, f, eX)is a Galois (resp. dual Galois) connec- tion.

(2) (eX, f, g, eY)is a Galois (resp. residuated) connection iff(e⁻¹_X , f, g, e⁻¹_Y )is a dual (resp. dual residuated) Galois connection.

(3) (eX, f, g, eY)is a residuated (resp. dual residuated) con- nection iff (e⁻¹_Y , g, f, e⁻¹_X ) is a residuated (resp. dual residuated) connection.

(4) (e_X, f, g, e_Y)is a Galois (resp. dual Galois) connection iff(eX, f, g, e⁻¹_Y )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 andf :X →Y andg:Y →Xdenote maps.

(1) (e_X, f, g, e_Y)is a Galois connection iff, gare antitone maps andeY(y, f(g(y))) =eX(x, g(f(x))) = 1.

(2) (eX, f, g, eY)is a dual Galois connection iff, gare anti- tone maps andeY(f(g(y)), y) =eX(g(f(x)), x) = 1.

(3) (eX, f, g, eY)is a residuated connection iff, gare iso- tone maps andeY(f(g(y)), y) =eX(x, g(f(x))) = 1.

(4) (eX, f, g, eY)is a dual residuated connection iff, gare isotone maps andeY(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 =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,

e_Y(f(x₁), f(x₂)) =e_X(x₂, g(f(x₁)))

≥e_X(x₂, x₁)e_X(x₁, g(f(x₁)))

=e_X(x₂, x₁).

e_X(g(y₁), g(y₂)) =e_Y(y₂, f(g(y₁)))

≥e_Y(y₂, y₁)e_Y(y₁, f(g(y₁)))

=e_Y(y₂, y₁).

Conversely,

e_Y(y, f(x))≥e_Y(y, f(g(y))e_Y(f(g(y)), f(x))

=e_Y(f(g(y)), f(x))

≥e_X(x, g(y)).

Similarly,e_Y(y, f(x))≤e_X(x, g(y)).

(2) SinceeY(f(x), y) =eX(g(y), x), we have 1 =e_Y(f(x), f(x)) =e_X(g(f(x)), x) and

eY(f(g(y)), y) =eX(g(y), g(y)) = 1.

Furthermore,

eY(f(x1), f(x2)) =eX(g(f(x2)), x1)

≥eX(x2, x1)eX(g(f(x2)), x2)

=eX(x2, x1).

eX(g(y1), g(y2)) =eX(f(g(y2)), y1)

≥eY(y2, y1)eY(f(g(y2)), y2)

=eY(y2, y1).

Conversely,

eY(f(x), y)≥eY(f(x), f(g(y))eY(f(g(y)), y)

=eY(f(x), f(g(y)))

≥eX(g(y), x).

Similarly,eY(f(x), y)≤eX(g(y), x).

(3) SinceeY(f(x), y) =eX(x, g(y)), we have 1 =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,

e_Y(f(x₁), f(x₂)) =e_X(x₁, g(f(x₂)))

≥e_X(x₁, x₂)e_X(x₂, g(f(x₂)))

=e_X(x₁, x₂).

e_X(g(y₁), g(y₂)) =e_X(f(g(y₁)), y₂)

≥e_Y(y₁, y₂)e_Y(f(g(y₁)), y₁)

=e_Y(y₁, y₂).

Conversely,

e_Y(f(x), y)≥e_Y(f(x), f(g(y))e_Y(f(g(y)), y)

=e_Y(f(x), f(g(y)))

≥e_X(x, g(y)).

Moreover,

eX(x, g(y))≥eX(x, g(f(x)))eX(g(f(x)), g(y))

=eX(g(f(x)), g(y))

≥eY(f(x), y).

(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]using

xy= max{0, x+y−1}, x→y= min{1−x+y,1}.

(1) Let(X={a, b, c}, e1)denote a fuzzy poset as follows:

e₁=











a b c

a 1.0 0.6 0.5 b 0.6 1.0 0.5 c 0.7 0.7 1.0











Sincee₁(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. Since0.7 =e1(c, a)6≤e1(f(a), f(c)) = e₁(b, c) = 0.5,fis not an antitone map. Hence,(e₁, f, f) are neither Galois nor dual Galois connections.

(2) Let(X={a, b, c}, e2)denote a fuzzy poset as follows:

e₂=











a b c

a 1.0 0.6 0.5 b 0.6 1.0 0.7 c 0.7 0.5 1.0











Sincee₂(x, y) =e₂(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 con-

nections. Since 0.7 = e2(c, a) 6≤ e2(f(c), f(a)) = e2(c, b) = 0.5,fis not an isotone map. Hence,(e2, f, f) are neither residuated nor dual residuated connections.

Definition 2.5. LetR ∈L^X×Y denote a fuzzy relation. For eachA∈L^XandB ∈L^Y, we define operationsR⁻¹(y, x) = R(x, y)andF,GR,HR,IR,JR,KR:L^Y →L^Xas follows:

FR(B)(x) = ^

y∈Y

(R(x, y)→B(y)),

GR(B)(x) = ^

y∈Y

(B(y)→R(x, y)), H_R(B)(x) = _

y∈V

(R(x, y)B(y)), IR(B)(x) = _

y∈V

(R^∗(x, y)B^∗(y)), JR(B)(x) = ^

y∈Y

(R(x, y)→B^∗(y)), K_R(B)(x) = _

y∈V

(R(x, y)B^∗(y)).

Theorem 2.6. LetR ∈ L^X×Y denote a fuzzy relation. For eachA∈L^XandB∈L^Y,

(1) HR(B) = (FR(B^∗))^∗,IR(B) = (G(B^∗))^∗.

2) GR(B) =FR^∗(B^∗),(IR(B))^∗=FR^∗(B) =GR(B^∗).

(3) H_R−1(FR(B))≤BandA≤ FR(H_R−1(A)).

(4) B ≤ FR⁻¹(HR(B))andHR(FR⁻¹(A))≤A.

(5) A≤ GR(G_R−1(A))andB ≤ G_R−1(GR(B)).

(6) A≤ JR(JR⁻¹(A))andB≤ JR⁻¹(JR(B)).

(7) IR(I_R⁻¹(A))≤AandI_R⁻¹(IR(B))≤B.

(8) K_R(K_R−1(A))≤AandK_R−1(K_R(B))≤B.

(9) F,HR: (L^Y, e_LY)→(L^X, e_LX)are isotone maps.

(10) G_R,I_R,J_R,K_R: (L^Y, e_LY)→(L^X, e_LX)are antitone maps.

Proof.(1) From Lemma 1.3 (13,14), we have (FR(B^∗))^∗(x)

= V

y∈X(R(x, y)→B^∗(y))^∗

= V

y∈X(R(x, y)B(y))^∗∗

=W

y∈X(R(x, y)B(y)) =H_R(B)(x).

(GR(B^∗))^∗(x)

= V

y∈X(B^∗(y)→R(x, y))∗

= V

y∈X(R^∗(x, y)B^∗(y))^∗∗

=W

y∈X(R^∗(x, y)B^∗(y)) =IR(B)(x).

(2)

FR^∗(B^∗)(x)

=V

y∈X(R^∗(x, y)→B^∗(y))(by Lemma 1.3 (12))

=V

y∈X(B(y)→R(x, y)) =GR(B)(x).

(IR(B))^∗(x)

= W

y∈X(R^∗(x, y)B^∗(y)∗

=V

y∈X(R^∗(x, y)→B(y)) =FR^∗(B)(x)

=V

y∈X(B^∗(y)→R(x, y)) =GR(B^∗)(x).

(3) From Lemma 1.3 (7), we have H_R−1(F_R(B))(x)

=W

y∈X(R⁻¹(x, y) FR(B)(y))

=W

y∈X(R(y, x)V

z∈X(R(y, z)→B(z))

≤W

y∈X(R(y, x)(R(y, x)→B(x))≤B(x).

From Lemma 1.3 (8), we have F_R(H_R−1(A))(x)

=V

y∈X(R(x, y)→ H_R⁻¹(A)(y))

=V

y∈X(R(x, y)→W

z∈X(R⁻¹(y, z)→A(z))

≥V

y∈X(R(x, y)→(R(x, y)A(x))≥A(x)

(4) From Lemma 1.3 (8), we have GR(GR⁻¹(A))(x)

=V

y∈X(G_R⁻¹(A)(y)→R(x, y))

=V

y∈X(V

z∈X(A(z)→R⁻¹(y, z))→R(x, y))

≥V

y∈X((A(x)→R(x, y))→R(x, y))

≥A(x).

(6)

JR(J_R−1(A))(x)

=V

y∈Y(R(x, y)→(JR⁻¹(A))^∗(y))

=V

y∈Y(R(x, y)→W

z∈X(R^∗−1(y, z)A(z)))

≥V

y∈Y(R(x, y)→(R(x, y)A(x)))

≥A(x).

(7)

IR(I_R⁻¹(A))(x)

=W

y∈Y(R^∗(x, y)(I_R−1(A))^∗(y))

=W

y∈Y(R^∗(x, y)V

z∈X(R^∗−1(y, z)→A(z)))

≤V

y∈Y(R^∗(x, y)(R^∗−1(y, x)→A(x)))

≤A(x).

(9)e_LY(A, B)≤e_LX(FR(A),FR(B))from:

F_R(A)(x)→ F_R(B)(x)

=V

y∈Y(R(x, y)→A(y))→V

y∈Y(R(x, y)→B(y))

≥V

y∈Y((R(x, y)→A(y))→(R(x, y)→B(y)))

≥V

y∈Y(A(y)→B(y)).(by Lemma 1.3 (7)) e_LY(A, B)≤e_LX(HR(A),HR(B))from:

HR(A)(x)→ HR(B)(x)

=W

y∈Y(R(x, y)A(y))→W

y∈Y(R(x, y)B(y))

≥V

y∈Y((R(x, y)A(y))→(R(x, y)B(y)))

≥V

y∈Y(A(y)→B(y)).(by Lemma 1.3 (9)) (10)e_LY(A, B)≤e_LX(G_R(B),G_R(A))from:

GR(B)(x)→ GR(A)(x)

=V

y∈Y(B(y)→R(x, y))→V

y∈Y(A(y)→R(x, y))

≥V

y∈Y((B(y)→R(x, y))→(A(y)→R(x, y)))

≥V

y∈Y(A(y)→B(y)).(by Lemma 1.3 (7)) e_LY(A, B)≤e_LX(I_R(B),I_R(A))from:

IR(B)(x)→ IR(A)(x)

=W

y∈Y(R^∗(x, y)B^∗(y))→W

y∈Y(R^∗(x, y)A^∗(y))

≥V

y∈Y((R^∗(x, y)B^∗(y))→(R^∗(x, y)A^∗(y)))

≥V

y∈Y(B^∗(y)→A^∗(y))(by Lemma 1.3 (9))

≥V

y∈Y(A(y)→B(y)).(by Lemma 1.3 (12)) Similarly,e_LY(A, B)≤e_LX(KR(B),KR(A)).

e_LY(A, B)≤e_LX(JR(B),JR(A))from:

JR(B)(x)→ JR(A)(x)

=V

y∈Y(R(x, y)→B^∗(y))→V

y∈Y(R(x, y)→A^∗(y))

≥V

y∈Y((R(x, y)→B^∗(y))→(R(x, y)→A^∗(y)))

≥V

y∈Y(B^∗(y)→A^∗(y))(by Lemma 1.3 (9))

≥V

y∈Y(A(y)→B(y)).(by Lemma 1.3 (7)) Other cases are similarly proved.

Theorem 2.7. LetR ∈L^X×Y denote a fuzzy relation,(L^X,

e_LX)and(L^Y, e_LY)denote fuzzy posets. We have the follow- ing properties.

(1) (e_LY,HR,F_R⁻¹, e_LX)and (e_LX,H_R⁻¹,FR, e_LY)are residuated connections.

(2) (e_LY,F_R,H_R−1, e_LX)and (e_LX,F_R−1,H_R, e_LY)are dual residuated connections.

(3) (e_LY,GR,G_R⁻¹, e_LX)and(e_LY,JR,J_R⁻¹, e_LX)are Ga- lois connections.

(4) (e_LY,IR,IR⁻¹, e_LX)and(e_LY,KR,KR⁻¹, e_LX)are dual Galois connections.

Proof.(1) For eachC∈L^X, B∈L^Y, e_LX(HR(B), C)

=V

x∈X(HR(B)(x)→C(x))

=V

x∈X

W

y∈Y(R(x, y)B(y))→C(x)

=V

x∈X

V

y∈V

B(y)→(R(x, y)→C(x))

=V

y∈Y

B(y)→V

x∈X(R⁻¹(y, x)→C(x))

=V

y∈Y

B(y)→ F_R⁻¹(C)(y)

=e_LY(B,FR⁻¹(C)).

(2) For eachC∈L^X, B∈L^Y, e_LX(C,FR(B))

=V

x∈X(C(x)→ FR(B)(x))

=V

x∈X

C(x)→V

y∈Y(R(x, y)→B(y))

=V

y∈Y

V

x∈X

(C(x)R(x, y))→B(y)

=V

y∈Y

W

x∈X(C(x)R(x, y))→B(y)

=V

y∈Y

H_R−1(C)(y)→B(y)

=e_LY(H_R−1(C), B).

(3) For eachC∈L^X, B∈L^Y, e_LX(C,JR(B))

=V

x∈X(C(x)→ JR(B)(x))

=V

x∈X

C(x)→V

y∈Y(R(x, y)→B^∗(y))

=V

x∈X

C(x)→V

y∈Y(R(x, y)B(y)→0)

=V

y∈Y

V

x∈X

(C(x)R(x, y)B(y))→0

=V

y∈Y

B(y)→V

x∈X((R(x, y)C(x))→0)

=V

y∈Y

B(y)→V

x∈X(R⁻¹(y, x)→C^∗(x)

=V

y∈Y

B(y)→ JR⁻¹(C)(y)

=e_LY(B,J_R−1(C)).

(4) For eachC∈L^X, B∈L^Y, e_LX(KR(B), C)

=V

x∈X(KR(B)(x)→C(x))

=V

x∈X

W

y∈Y R(x, y)B^∗(y)→C(x)

=V

x∈X

V

y∈Y

R(x, y)B^∗(y)C^∗(x)→0

=V

x∈X

V

y∈Y

R(x, y)C^∗(x)→B(y)

=V

y∈Y

W

x∈X(R⁻¹(y, x)C^∗(x))→B(y)

=V

y∈Y

K_R⁻¹(C)(y)→B(y)

=e_LY(KR⁻¹(C), B).

For eachC∈L^X, B∈L^Y, e_LY(IR(B), C)

=V

x∈X(IR(B)(x)→C(x))

=V

x∈X

W

y∈Y(R^∗(x, y)B^∗(y)→C(x)

=V

x∈X

V

y∈Y

R^∗(x, y)B^∗(y)→C(x)

=V

x∈X

V

y∈Y

R^∗(x, y)C^∗(x)→B(y)

=V

y∈Y

W

x∈X((R⁻¹)^∗(y, x)C^∗(x))→B(y)

=V

y∈Y

I_R−1(C)(y)→B(y)

=e_LY(I_R−1(C), B).

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 latticeL. In particular, we studied fuzzy Galois (dual Galois, residuated, and dual residuated) connec- tions 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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Yong Chan Kimreceived the B.S., M.S., and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1982, 1984, and 1991, respectively. He is cur- rently Professor of Gangneung-Wonju Uni- versity. His research interests include fuzzy topology and fuzzy logic.

E-mail: yck@gwnu.ac.kr