Galois and Residuated Connections on Sets and Power Sets

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1 Galois and Residuated Connections on Sets and Power Sets Yong Chan Kim 1 and Jung Mi Ko 2 Department of Mathematics, Gangneung-Wonju National University, Gangneung, , Korea Abstract We investigate the relations between various connections on set and those on power set. Moreover, we give their examples. Key Words: Galois, dual Galois, residuated and dual residuated connections 1. Introduction and Preliminaries An information consists of X, Y, R where X is a set of objects, Y is a set of attributes and R is a relation between X and Y. Wille [11] introduced the formal concept lattices by allowing some uncertainty in data as examples as Galois, dual Galois, residuated and dual residuated connections. Formal concept analysis is an important mathematical tool for data analysis and knowledge processing [1-5,8,9,11]. In this paper, we show that Galois, dual Galois, residuated and dual residuated connections on set induce various connections on power sets. In particular, we give their examples. Let X be a set. A relation e X X X is called a partially order set simply, posetif it is reflexive, transitive and anti-symmetric. We can define a poset e P X P X P X as A, B e P X iff A B for A, B P X. If X, e X is a poset and we define a function x, y e 1 X iff y, x e X, then X, e 1 X is a poset. Definition 1.1. [10] Let X, e X and Y,e Y be posets and f : X Y and g : Y X maps. 1 e X,f,g,e Y is called a Galois connection if for all x X, y Y, y, fx e Y iff x, gy e X. 2e X,f,g,e Y is called a dual Galois connection if for all x X, y Y, fx,y e Y iff gy,x e Y. 3 e X,f,g,e Y is called a residuated connection if for all x X, y Y, fx,y e Y iff x, gy e Y. 4 e X,f,g,e Y is called a dual residuated connection if for all x X, y Y, y, fx e Y iff gy,x e Y. 5 f is an isotone map if fx 1,fx 2 e Y for all x 1,x 2 e X. 6 f is an antitone map if fx 2,fx 1 e Y for all x 1,x 2 e X. Theorem 1.2. [10] Let X, e X and Y,e Y be a poset and f : X Y and g : Y X maps. 1 e X,f,g,e Y is a Galois connection if f,g are antitone maps and y, fgy e Y and x, gfx e X. Manuscript received Nov. 14, 2010; revised May. 27, e X,f,g,e Y is a dual Galois connection if f,g are antitone maps and fgy,y e Y and gfx,x e X. 3 e X,f,g,e Y is a residuated connection if f,g are isotone maps and fgy,y e Y and x, gfx e X. 4 e X,f,g,e Y is called a dual residuated connection if f,g are isotone maps and y, fgy e Y and gfx,x e X. Theorem 1.3. [9] 1 e P X,F,G,e P Y is a Galois connection iff F i Γ A i= i Γ F A i. 2 e P X,F,G,e P Y is a residuated connection iff F i Γ A i= i Γ F A i. 3 e P X,F,G,e P Y is a dual Galois connection iff F i Γ A i= i Γ F A i. 4 e P X,F,G,e P Y is a dual residuated connection iff F i Γ A i= i Γ F A i. 2. Galois and Residuated Connections on Sets and Power Sets Theorem 2.1. Let X, e X and Y,e Y be a poset and f : X Y and g : Y X maps. For each A P X and B P Y, we define operations as follows: F 1 A ={y Y x Xx A fx,y e Y }, F 2 A ={y Y x Xx A y, fx e Y }, G 1 B ={x X y Y y B x, gy e X }, G 2 B ={x X y Y y B gy,x e X }, H 1 B ={x X y Y y B &x, gy e X }, H 2 B ={x X y Y y B &gy,x e X }, I 1 A ={y Y x Xx A &y, fx e Y }, I 2 A ={y Y x Xx A &fx,y e Y },

2 J 1 B ={x X y Y x, gy e X y B}, J 2 B ={x X y Y gy,x e X y B}, K 1 A ={y Y x Xfx,y e Y x A}, K 2 A ={y Y x Xy, fx e Y x A}, L 1 B ={x X y Y y B c &x, gy e X }, L 2 B ={x X y Y y B c &gy,x e X }, M 1 A ={y Y x Xx A c &y, fx e Y }, M 2 A ={y Y x Xx A c &fx,y e Y }, Then the following statements hold: 1 F 1 {x} =I 2 {x} =M 2 {x} c =e Y fx, F 2 {x} = I 1 {x} = M 1 {x} c = e Y 1 fx, K 2 {x} c =e Y 1 fx c, K 1 {x} c =e Y c fx. 2 G 1 {y} = H 1 {y} = L 1 {y} c = e X 1 gy, G 2 {y} = H 2 {y} = L 2 {y} c = e X gy. J 1 {y} c =e X 1 gy c, J 2 {y} c =e X c gy. 3 e X,f,g,e Y is a Galois connection iff e P X,F 2,G 1,e P Y is a Galois connection iff e P X,K 2,H 1,e P Y is a dual residuated connection iff e P X,M 1,L 1,e P Y is a dual Galois connection iff e P X,I 1,J 1,e P Y is a residuated connection. 4 e X,f,g,e Y is a residuated connection iff e P X,F 1,G 1,e P Y is a Galois connection iff e P X,K 1,H 1,e P Y is a dual residuated connection iff e P X,M 2,L 1,e P Y is a dual Galois connection iff e P X,I 2,J 1,e P Y is a residuated connection. 5 e X,f,g,e Y is a dual Galois connection iff e P X,F 1,G 2,e P Y is a Galois connection iff e P X,K 1,H 2,e P Y is a dual residuated connection iff e P X,M 2,L 2,e P Y is a dual Galois connection iff e P X,I 2,J 2,e P Y is a residuated connection. 6 e X,f,g,e Y is a dual residuated connection iff e P X,F 2,G 2,e P Y is a Galois connection iff e P X,K 2,H 2,e P Y is a dual residuated connection iff e P X,M 1,L 2,e P Y is a dual Galois connection iff e P X,I 1,J 2,e P Y is a residuated connection. 7 If f i Γ x i = i Γ fx i for x i X, there exists a function g : Y X such that e X,f,g,e Y is a Galois connection. Moreover, e P X,F 2,G 1,e P Y is a Galois connection, e P X,K 2,H 1,e P Y is a dual residuated connection, e P X,M 1,L 1,e P Y is a dual Galois connection and e P X,I 1,J 1,e P Y is a residuated connection. 8 If f i Γ x i = i Γ fx i for x i X, there exists a function g : Y X such that a residuated connection. Moreover, e P X,F 1,G 1,e P Y is a Galois connection, e P X,K 1,H 1,e P Y is a dual residuated connection, e P X,M 2,L 1,e P Y is a dual Galois connection and e P X,I 2,J 1,e P Y is a residuated connection. 9 If f i Γ x i = i Γ fx i for x i X, there exists a function g : Y X such that e X,f,g,e Y is a dual Galois connection. Moreover, e P X,F 1,G 2,e P Y is a Galois connection, e P X,K 1,H 2,e P Y is a dual residuated connection, e P X,M 2,L 2,e P Y is a dual Galois connection and e P X,I 2,J 2,e P Y is a residuated connection. 10 If f i Γ x i = i Γ fx i for x i X, there exists a function g : Y X such that e X,f,g,e Y is a dual residuated connection. Moreover, e P X,F 2,G 2,e P Y is a Galois connection and e P X,K 2,H 2,e P Y is a dual residuated connection, e P X,M 1,L 2,e P Y is a dual Galois connection and e P X,I 1,J 2,e P Y is a residuated connection. Proof. 1 y F 1 {x} iff z Xz {x} fz,y e Y iff fx,y e Y. Similarly, F 2 {x} = e Y 1 fx. We K 1{x} c = e Y c fx from: y K 1 {x} c iff z Xfz,y e Y z {x} c iff z Xz {x} fz,y e Y iff fx,y e Y. Similarly, K 2 {x} c =e Y 1 fx c. Other cases and 2 are similarly proved as in 1. 3 First, if x, gy e X iff y, fx e Y, then A, G 1 B e P X iff B,F 2 A e P Y. B,F 2 A e P Y iff y Y y B y F 2 A iff y Y y B x Xx A y, fx e Y iff y Y x X x A y B x, gy e X iff x X x A y Y y B x, gy e X iff x X x A x G 1 B iff A, G 1 B e P X. Conversely, put A = {x} and B = {y}. Since F 2 {x} =e Y 1 fx and G 1{y} =e X 1 gy from 1 and 2, we y, fx e Y iff {y},f 2 {x} e P Y iff {x},g 1 {y} e P X iff x, gy e X. Second, if x, gy e X iff y, fx e Y, then H 1 B,A e P X iff B,K 2 A e P Y.

3 I 1 A,B e P Y iff A, J 1 B e P X. H 1 B,A e P X iff x Xx H 1 B x A iff x X y Y x, gy e X & y B x A iff x X y Y y B x, gy e X x A iff y Y y B x Xy, fx e Y x A iff y Y y B y K 2 A iff B,K 2 A e P X. Put A = {x} c and B = {y}. Since K 2 {x} c = e Y 1 fx c and H 1 {y} = e X 1 gy from 1 and 2, we x, gy e X iff H 1 {y}, {x} c e P X iff {y},k 2 {x} c e P Y iff y, fx e Y. Third, if x, gy e X iff y, fx e Y, then L 1 B,A e P X iff M 1 A,B e P Y. M 1 A,B e P Y iff y Y y M 1 A y B iff y Y z Xz A c &y, fz e Y y B iff y Y z Xz A c y, fz e Y y B iff z Xz A c y Y y, fz R y B iff z X y Y z,gy e X & y B c z A iff L 1 B,A e P X. Put A = {x} c and B = {y} c. Since M 1 {x} c = e Y 1 fx and L 1{y} c =e X 1 gy from 1 and 2, we I 1 A,B e P Y iff y Y y I 1 A y B iff y Y x Xx A &y, fx e Y y B iff y Y x Xx A y, fx e Y y B iff x Xx A y Y x, gy e X y B iff x Xx A x J 1 B iff A, J 1 B e P X. Put A = {x} and B = {y} c. Since I 1 {x} = e Y 1 fx and J 1{y} c = e X 1 gy c from 1 and 2, we y, fx e Y iff y I 1 {x} c iff I 1 {x}, {y} c e P Y iff {x},j 1 {y} c e P X iff x J 1 {y} c iff x, gy e X. 4 Let x, gy e X iff fx,y e Y be given. Then A, G 1 B e P X iff B,F 1 A e P Y from: B,F 1 A e P Y iff y Y y B y F 1 A iff y Y y B x Xx A fx,y e Y iff y Y x X x A y B x, gy e X iff x X x A y Y y B x, gy e X iff x X x A x G 1 B iff A, G 1 B e P X. Conversely, put A = {x} and B = {y}. Since F 1 {x} = e Y fx and G 1 {y} =e X 1 gy from 1 and 2, we fx,y e Y iff y F 1 {x} y, fx e Y iff y M 1 {x} c c iff M 1 {x} c, {y} c e P Y iff L 1 {y} c, {x} c e P X iff x L 1 {y} c c iff x, gy e X. iff {y},f 1 {x} e P Y iff {x},g 1 {y} e P X iff x G 1 {x} iff x, gy e X. Forth, if x, gy e X iff y, fx e Y, then Second, if x, gy e X iff fx,y e Y, then

4 H 1 B,A e P X iff B,K 1 A e P Y from: H 1 B,A e P X iff x Xx H 1 B x A iff x X y Y x, gy e X & y B x A iff x X y Y y B x, gy e X x A iff y Y y B x Xfx,y e Y x A iff y Y y B y K 1 A iff B,K 1 A e P Y. Put A = {x} c and B = {y}. Since K 1 {x} c = e Y c fx and H 1{y} =e X 1 gy from 1 and 2, we x, gy e X iff x H 1 {y} iff H 1 {y}, {x} c e P X iff {y},k 1 {x} c e P Y iff y K 1 {x} iff fx,y e Y. Third, if x, gy e X iff fx,y e Y, then L 1 B,A e P X iff M 2 A,B e P Y. M 2 A,B e P Y iff y Y y M 2 A y B iff y Y z Xz A c &fz,y e Y y B iff y Y z Xz A c fz,y e Y y B iff z Xz A c y Y fz,y R y B iff z X y Y z,gy e X & y B c z A iff L 1 B,A e P X. Put A = {x} c and B = {y} c. Since M 2 {x} c = e Y fx and L 1 {y} c =e X 1 gy from 1 and 2, we fx,y e Y iff y M 2 {x} c c iff M 2 {x} c, {y} c e P Y iff L 1 {y} c, {x} c e P X iff x L 1 {y} c c iff x, gy e X. Forth, if x, gy e X iff fx,y e Y, then I 2 A,B e P Y iff A, J 1 B e P X. I 2 A,B e P Y iff y Y y I 2 A y B iff y Y x Xx A &fx,y e Y y B iff y Y x Xx A fx,y e Y y B iff x Xx A y Y x, gy e X y B iff x Xx A x J 1 B iff A, J 1 B e P X. Put A = {x} and B = {y} c. Since I 2 {x} = e Y fx and J 1 {y} c = e X 1 gy c from 1 and 2, we fx,y e Y iff y I 1 {x} c iff I 1 {x}, {y} c e P Y iff {x},j 1 {y} c e P X iff x J 1 {y} c iff x, gy e X. 5 and 6 are similarly proved as in 3 and 4. 7 Define gy = {x y fx}. Ify fx, then x gy. Ifx gy, then fx fgy = f x = fx y. Hence e X,f,g,e Y is a Galois connection. Others follow 3. 8 Define gy = {x fx y}. If fx y, then x gy. If x gy, then fx fgy = f x = fx y. Hence e X,f,g,e Y is a residuated connection. Others follow 4. 9 Define gy = {x fx y}. Iffx y, then gy x. Ifgy x, then fx fgy = f x = fx y. Hence e X,f,g,e Y is a dual Galois connection. Others follow Define gy = {x y fx}. If y fx, then gy x. If gy x, then fx fgy = f x = fx y. Hence e X,f,g,e Y is a dual residuated connection. Others follow 6. Example 2.2. Let X = {a, b, c, d},e X and Y = {x, y, z},e Y be a poset with e X = {a, a, a, b, a, c, a, d, b, b, b, d, c, c, c, d, d, d}. e Y = {x, x, x, y, x, z, y, y, z,y, z,z}. 1 Let f : X Y as fa =x, fb =z,fc = fd =y. Then f i Γ x i = i Γ fx i for x i X. Define gw = {s X fs w}. Then g : Y X as gx =a, gy =d, gz =b. Then e X,f,g,e Y is a residuated connection. By Theorem and 2, we obtain F 1 = I 2 {a} =M 2 {a} c ={x, y, z} =e Y fa, F 1 = I 2 {b} =M 2 {a} c ={y, z} =e Y fb, F 1 = I 2 {c} =M 2 {a} c ={y} =e Y fc, F 1 = I 2 {d} =M 2 {a} c ={y} =e Y fd. K 1 {a} c = =e Y c fa, K 1 {b} c = {x} =e Y c fb, K 1 {c} c = {x, z} =e Y c fc, K 1 {d} c = {x, z} =e Y c fd.

5 G 1 = H 1 {x} =L 1 {x} c ={a} =e X 1 gx, G 1 = H 1 {y} =L 1 {y} c =X =e Y 1 gy, G 1 = H 1 {z} =L 1 {z} c ={a, b} =e Y 1 gz. J 1 {x} c ={b, c, d} =e X 1 gx c, J 1 {y} c = =e Y 1 gy c, J 1 {z} c ={c, d} =e Y 1 gz c. We obtain F 1 : P X P Y and G 1 : P Y P X as follows: Y if A {, {a}}, F 1 A = {y, z} if A {{b}, {a, b}}, {y} otherwise, X G 1 B = {a, b} {a} if B {, {y}}, if B {{z}, {y, z}}, otherwise. Then e P X,F 1,G 1,e P Y is a Galois connection. We obtain K 1 : P X P Y and H 1 : P Y P X as follows: K 1 A = H 1 B = Y if A = X, {x} if a A, {a, b} A, {x, z} if {a, b} A X, otherwise, if B =, X if y B, {a, b} if z B,y B, {a} otherwise. Then e P X,K 1,H 1,e P Y is a dual residuated connection. We obtain M 2 : P X P Y and L 1 : P Y P X as follows: if A = X, {y} if A {{a, b}, {a, b, c}, {a, b, d}}, M 2 A = {y, z} if A {{a}, {a, d}, {a, c, d}}, if B = X, {a} if B = {y, z}, L 1 B = {a, b} if B {{y}, {x, y}, X otherwise. Then e P X,M 2,L 1,e P Y is a dual Galois connection. We obtain I 2 : P X P Y and J 1 : P Y P X as follows: if A =, {z} if A {{c}, {d}, {c, d}}, I 2 A = {y, z} if A {{b}, {b, c}, {b, d}, {b, c, d}}, X if B = Y, {b, c, d} if B = {y, z}, J 1 B = {c, d} if B {{y}, {x, y}}, otherwise. Then e P X,I 2,J 1,e P Y is a residuated connection. 2 Let f : X Y as fa =y, fb =z,fc = fd =x. Then f i Γ x i = i Γ fx i for x i X. Define gw = {s X fs w}. Then g : Y X as gx =d, gy =a, gz =b. Then e X,f,g,e Y is a Galois connection. By Theorem and 2, we obtain F 2 = I 1 {a} =M 1 {a} c =Y =e Y 1 fa, F 2 = I 1 {b} =M 1 {b} c ={x, z} =e Y 1 F 2 = I 1 {c} =M 1 {c} c ={x} =e Y 1 fc, F 2 = I 1 {d} =M 1 {d} c ={x} =e Y 1 fb, fd. G 1 = H 1 {x} =L 1 {x} c =X =e X 1 gx, G 1 = H 1 {y} =L 1 {y} c ={a} =e Y 1 gy, G 1 = H 1 {z} =L 1 {z} c ={a, b} =e Y 1 J 1 {x} c = =e X 1 gx c, J 1 {y} c ={b, c, d} =e Y 1 J 1 {z} c ={c, d} =e Y 1 K 2 {a} c K 2 {b} c K 2 {c} c K 2 {d} c = =e 1 = {y} =e 1 = {y, z} =e 1 = {y, z} =e 1 gy c, gz c. Y c fa, Y c fb, Y c fc, Y c fd. gz. We obtain F 2 : P X P Y and G 1 : P Y P X as follows: Y if A {, {a}}, F 2 A = {x, z} if A {{b}, {a, b}}, {x} otherwise, X if B {, {x}}, G 1 B = {a, b} if B {{z}, {x, z}}, {a} otherwise. Then e P X,F 2,G 1,e P Y is a Galois connection. We obtain K 2 : P X P Y and H 1 : P Y P X as follows: Y if A = X, {y} if a A, {a, b} A, K 2 A = {y, z} if {a, b} A X, otherwise, H 1 B = if B =, X if x B, {a, b} if z B,x B, {a} otherwise. Then e P X,K 2,H 1,e P Y is a dual residuated connection.

6 We obtain M 1 : P X P Y and L 1 : P Y P X as follows: if A = X, {x} if A {{a, b}, {a, b, c}, {a, b, d}}, M 1 A = {x, z} if A {{a}, {a, c}, {a, d}, {a, c, d}}, if B = Y, {a} if B = {x, z}, L 1 B = {a, b} if B {{x}, {x, y}}, X otherwise. Then e P X,M 1,L 1,e P Y is a dual Galois connection. We obtain I 1 : P X P Y and J 1 : P Y P X as follows: if A =, {z} if A {{c}, {d}, {c, d}}, I 1 A = {x, z} if A {{b}, {b, c}, {b, d}, {b, c, d}}, X if B = Y, {b, c, d} if B = {x, z}, J 1 B = {b, d} if B {{x}, {x, y}}, otherwise. Then e P X,I 1,J 1,e P Y is a residuated connection. References [1] R. Bělohlávek, Similarity relations in concept lattices, J. Logic and Computation, vol. 10, no 6,pp , [2] R. Bělohlávek, Lattices of fixed points of Galois connections, Math. Logic Quart., vol. 47, pp , [3] R. Bělohlávek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, vol. 128, pp , [5] G. Georgescu, A. Popescue, Non-dual fuzzy connections, Arch. Math. Log., vol. 43, pp , [6] J.G. Garcia, I.M. Perez, M.A.P. Vicente, D. Zhang, Fuzzy Galois connections categorically, Math. Log. Quart., vol. 56, pp , [7] H. Lai, D. Zhang, Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory, Int. J. Approx. Reasoning, vol. 50, pp , [8] Y.C. Kim, J.W. Park, Join preserving maps and various concepts, Int.J. Contemp. Math. Sciences, vol. 5, no.5, pp , [9] J.M. Ko, Y.C. Kim, Various connections and their relations, submit to Int. J. Fuzzy Logic and Intelligent Systems. [10] Ewa. Orlowska, I. Rewitzky, Algebras for Galoisstyle connections and their discrete duality, Fuzzy Sets and Systems, vol.161, pp , [11] R. Wille, Restructuring lattice theory; an approach based on hierarchies of concept, in: 1. RivalEd., Ordered Sets, Reidel, Dordrecht, Boston, [12] Wei Yao, Ling-Xia Lu, Fuzzy Galois connections on fuzzy posets, Math. Log. Quart., vol. 55, pp , Yong Chan Kim He received the M.S and Ph.D. degrees in Department of Mathematics from Yonsei University, in 1984 and 1991, respectively. From 1991 to present, he is a professor in the Department of Mathematics, Gangneung -Wonju University. His research interests are fuzzy topology and fuzzy logic. yck@gwnu.ac.kr [4] R. Bělohlávek, Fuzzy relational systems, Kluwer Academic Publisher, New York, fuzzy logic. Jung Mi Ko She received the M.S and Ph.D. degrees in Department of Mathematics from Yonsei University, in 1983 and 1988, respectively. From 1988 to present, she is a professor in Department of Mathematics, Gangneung- Wonju University. Her research interests are jmko@gwnu.ac.kr