International Mathematical Forum, 4, 009, no., 89-99 Pseudo MV-Algebras Induced by Functions Yong Chan Kim Department of Mathematics, Kangnung National University, Gangneung, Gangwondo 10-70, Korea yck@kangnung.ac.kr Jin Won Park Department of Mathematics Education, Cheju National University, Jeju, Jejudo 690-756, Korea jinwon@cheju.ac.kr Abstract In this paper, we investigate the relationships between lattice-groups and two types of pseudo MV-algebras. Moreover, we find pseudo MValgebras induced by functions. We give their examples. Mathematics Subject Classification: 06D35, 06D50, 06F05, 06F15 Keywords: lattice-group, positive cone, pseudo MV-algebras 1 Introduction and preliminaries Noncommutative structures play an important role in metric spaces, algebraic structures (groups, rings, quantales, BL-algebras). Georgescu and Iorgulescu [6] introduced pseudo MV-algebras as the generalization of the MV-algebras of Chang []. Recently, they are developed many directions [3-5,7,8]. In this paper, we define the pseudo MV-algebra with respect to residuated lattice in a sense [9-11]. We prove that a lattice-group [1] with positive cone generates two types of pseudo MV-algebras. Moreover, we find pseudo MValgebras induced by functions. We give their examples. Definition 1.1 [1] A lattice (L,, ) is called a lattice-group iff it satisfies the following conditions: (L1) (L, ) is a group, (L) ( a ) b =( b) (a b), b ( a )=(b ) (b a ), for all,a,b L.
90 Yong Chan Kim and Jin Won Park In a lattice-group L with identity element e, P = {x L x e} is called a positive cone if it satisfies P P 1 = {e}, P P P, (a 1 ) P a = P a L where P 1 = {x L x 1 P }. Definition 1. [6,9-11] A structure (L,,,,, r, l,, ) is called a pseudo-mv algebra if it satisfies: (M1) (L,,,, ) is a bounded lattice where and are the greatest element and the least element, respectively. (M) (x y) z = x (y z) for all x, y, z L, (M3) x = x = x for all x X, (M4) z x r y iff x z y iff x z l y for all x, y, z L, (M5) x y =(x l y) x = x (x l y), (M6) (x l y) (y l x)=(x r y) (y r x)=, (M7) x = x = x where a = a r and a = a r. A pseudo-mv algebra is called a MV-algebra if it is commutative. Pseudo MV-algebras and lattice-groups Lemma.1 Let (L, ) be a group, P a positive cone and L = P P 1. Define a b iff b a 1 P and a 1 b P. Then (L,, ) is a lattice-group and (L, ) is a totally ordered set. Proof. (reflexive) Since x x 1 = x 1 x = e P, x x. (transitive) Let x y and y z. Then y x 1 P, x 1 y P, z y 1 P and y 1 z P.Thusx z because z y 1 y x 1 = z x 1 P x 1 y y 1 z = x 1 z P. (anti-symmetry) Let x y and y x. Then y x 1 P and x y 1 P. Hence y x 1 P P 1 = {e}. Sox = y. (comparable) For x, y L, since L = P P 1, x 1 y P or x 1 y P 1 ;i.e. x y or y x. Thus, (L, ) is a totally ordered set. Let x y. Then y x 1 P. (a y) (a x) 1 =(a y) x 1 a 1 = a (y x 1 ) a 1 P. Similarly, x 1 y P implies a (x 1 y) a 1 P. Hence a x a y. Moreover, a (x y) =(a x) (a y). Similarly, (x y) a =(x a) (y a). Thus (L,, ) is a lattice-group.
Pseudo MV-algebras induced by functions 91 Theorem. Let (L, ) be a lattice-group with identity e, P a positive cone, L [a, e] ={x L a x e} and L + [e, b] ={x L e x b}. Then: (1) (L 1 [a, e],, r, l,a,e) is a pseudo-mv-algebra where a is the least element and e is the greatest element and x y =(x y) a, x r y =(x 1 y) e, x l y =(y x 1 ) e. () (L + [e, b],, r, l,e,b) is a pseudo-mv-algebra where e is the least element and b is the greatest element and x y =(x b 1 y) e, x r y =(b x 1 y) b, x l y =(y x 1 b) b. Proof. (1) (M) If x y z e, then (x y) z = e = x (y z). If x y z e, then (x y) z = x y z = x (y z). If x 1 x, then (x y) (x 1 y) 1 = x x 1 1 P and (x y) 1 (x 1 y) =x 1 x 1 P. So, (x 1 y) (x y). Furthermore, (x 1 y) (x y). (M3) x e =(x e) a = x a = x. (M4) Let x y =(x y) a z. Then y (x 1 z) e = x r y. ( Let y (x 1 z) e = x r y. Then x y x ((x 1 z) e) = x ((x 1 z) e) ) a = z x z. (M5) x (x r y) = x ((x 1 y) e) = ( x ((x 1 y) e) ) a = ( (x x 1 y) (x e) ) a = y x. (M6) Since (x 1 y) P or (x 1 y) P 1, we have (x 1 y) e or (y 1 x) e. Thus, (x r y) (y r x)= ( (x 1 y) e ) ( (y 1 x) e ) = e. (M7) Since (x 1 a) x 1 x = e, we have (x r a) l a = ( (x 1 a) e ) l a =(x 1 a) l a = ( a (x 1 a) 1) e =((a a 1 ) x) e = x. Other cases are similarly proved. Hence it is a pseudo-mv-algebra. () (M1) and (M3) are easily proved.
9 Yong Chan Kim and Jin Won Park (M) If x y z e e, then (x y) z = e = x (y z). If x y z e e, then (x y) z = x y z = x (y z). If x 1 x, then (x y b 1 ) (x 1 y b 1 ) 1 = x x 1 1 P and (x 1 y b 1 ) 1 (x y b 1 )=b y 1 x 1 1 x y b 1 P because a P a 1 P. So, (x 1 y) (x y). Since e z e b = e and x e e, we have (x y) z = ( (x b 1 y) e ) z = ( (x b 1 y) z ) (e z) =(x b 1 y b 1 z) e. x (y z) = x ( (y b 1 z) e ) =(e z) ( x (y b 1 z) ) =(x b 1 y b 1 z) e. Hence (x y) z = x (y z). (M4) Let x y =(x b 1 y) e z. Then y (b x 1 z) b = x r z. Let y (b x 1 z) b = x r z. Then x y x b 1 (b x 1 z) b) =z x z. Similarly, x y z iff x y l z. (M5) x (x r y) = x ((b x 1 y) b) = ( x b 1 ((b x 1 y) b) ) e = ( (x b 1 b x 1 y) (x b 1 b) ) e = y x. Similarly, (x r y) x = x y. (M6) Since (b x 1 y b 1 ) P or (b x 1 y b 1 ) 1 = b y 1 x b 1 P, we have b x 1 y b or b y 1 x b. Thus, (x r y) (y r x)= ( (b x 1 y) b ) ( (b y 1 x) b ) = b. (M7) Since (b x 1 e) b x 1 x = b e = b, we have (x r e) l e = ( (b x 1 e) b ) l e =(b x 1 ) l e = ( e (b x 1 ) 1 b ) b =(x b 1 b) b = x. Hence it is a pseudo-mv-algebra.
Pseudo MV-algebras induced by functions 93 Example.3 Let L = {(x, y) R x, y R} be a set and we define an operation : L L L as follows: (x 1, ) (x,y )=(x 1 + x, + y ). Then (L, ) is a group with e =(0, 0) and (x, y) 1 =( x, y). Let P = {(a, b) R a =0,b 0ora>0}. We have P P 1 = {(0, 0)}, P P P,(a, b) 1 P (a, b) =P and P P 1 = R. Hence P is a positive cone. We define (x 1, ) (x,y ) (x 1, ) 1 (x,y ) P x 1 = x, y or x 1 <x. Then (L, ) is a lattice-group. (1) (L [( 3, ), (0, 0)],, r, l, ( 3, ), (0, 0)) is an MV-algebra where ( 3, ) is the least element and (0, 0) is the greatest element from the following statements: (x 1, ) (x,y ) = (x 1, ) (x,y ) ( 3, ) =(x 1 + x, + y ) ( 3, ). Since (x 1, ) (x,y )=(x,y ) (x 1, ), we have r = l as (x 1, ) r (x,y ) = ((x 1, ) 1 (x,y )) (0, 0) =(x x 1,y ) (0, 0). Furthermore, we have (x, y) =(x, y) =(x, y) from: (x, y) =(x, y) =( x 3, y ). If (x 1, ) (x,y ), then (x 1, ) 1 (x,y ) P. So, (x 1, ) l (x,y )= (0, 0). Hence ((x 1, ) l (x,y )) (x 1, )=(x 1, ). If (x 1, ) (x,y ), then (x 1, ) l (x,y )=(x,y ) (x 1, ) 1 (x 1, )=(x,y ). Hence ((x 1, ) l (x,y )) (x 1, )=(x 1, ) (x,y ). Since (x 1, ) 1 (x,y ) P and (x 1, ) 1 (x,y ) P 1, we have ((x 1, ) r (x,y )) ((x,y ) r (x 1, )) = (0, 0). () (L + [(0, 0), (, 3)],, r, l, (0, 1), (, 3)) is an MV-algebra where (0, 0) is the least element and (, 3) is the greatest element from the following statements: (x 1, ) (x,y ) = ( (x 1, ) (, 3) 1 (x,y ) ) (0, 0) =(x 1 + x, + y 3) (0, 0).
94 Yong Chan Kim and Jin Won Park Since (x 1, ) (x,y )=(x,y ) (x 1, ), we have r = l as (x 1, ) r (x,y ) = ((x 1, ) 1 (x,y )) (, 3) 1 (, 3) =( x 1 + x +, + y +3) (, 3). Furthermore, we have (x, y) =(x, y) =(x, y) from: (x, y) =(x, y) =(x, y) r (0, 0) = ( x +, y +3). Other cases are similarly proved as in (1). Example.4 Let L = {x R x>1} be a set and we define an operation : L L L as follows: x y = xy x y +. Then (L, ) is a group with e =,x 1 = x. x 1 We have a positive cone P = {x R x } because P P 1 = {}, P P P, x 1 P x = P and P P 1 = L. We define x 1 x (x 1 1 x ) P x 1 x 1 1 x = x 1 +1. x 1 1 Then (L, ) is a lattice-group. (1) (L [ 3, ],,, 3, ) is an MV-algebra. Define x y =(x y) 3 x y =(x 1 y) = x + y. x 1 We obtain x (x y) =x y. (a) if x y, then x (x y) =x. (b) if x y, then x (x y) =y. Furthermore, (x y) (y x) = and (x 3) 3 =. () (L + [, 4],,,, 4) is an MV-algebra. Define x y =(x 4 1 y) = 1 (xy x y +4), 3 x y =(4 x 1 y) 4 = (3( x+y ) ) 4. x 1 If x y, then x y 3( x ) 4. We obtain x (x y) =x y. x 1 Furthermore, (x y) (y x) = 4 and (x ) =x.
Pseudo MV-algebras induced by functions 95 Example.5 Let L = {(x, y) R y>0} be a set and we define an operation : L L L as follows: (x 1, ) (x,y )=(x 1 + x, y ). Then (L, ) is a group with e =(0, 1), (x, y) 1 =( x y, 1 y ). We have a positive cone P = {(a, b) R b =1,a 0, or y>1} because P P 1 = {(0, 1)}, P P P,(a, b) 1 P (a, b) =P and P P 1 = L. For (x 1, ), (x,y ) L, we define (x 1, ) (x,y ) (x 1, ) 1 (x,y ) P, (x,y ) (x 1, ) 1 P <y or = y,x 1 x. Then (L, ) is a lattice-group. (1) (L [(1, 1 ), (0, 1)],, r, l, (1, 1 ), (0, 1)) is a pseudo MV-algebra where (1, 1 ) is the least element and (0, 1) is the greatest element from the following statements: (x 1, ) (x,y ) = (x 1, ) (x,y ) (1, 1 )=(x 1 + x, y ) (1, 1 ), (x 1, ) r (x,y ) = ((x 1, ) 1 (x,y )) (0, 1) = ( x 1+x, y ) (0, 1), (x 1, ) l (x,y ) = ((x,y ) (x 1, ) 1 ) (0, 1) = (x x 1y, y ) (0, 1). It is not commutative because ( 3, 3 4 ) (4, 1 ) = (3 + 3, 3 8 ) (4, 1 ) ( 3, 3 4 ) = (4 + 1 3, 3 8 ). Furthermore, we have (x, y) =(x, y) =(x, y) from: (x, y) =(x, y) r (1, 1 +1 )=( x, 1 y y ), (x, y) =(x, y) l (1, 1 )=(1 x y, 1 y ). Other cases are proved as a similar method in Example.3(1). () (L + [(0, 1), (, 3)],, r, l, (0, 1), (, 3)) is a pseudo MV-algebra where (0, 1) is the least element and (, 3) is the greatest element from the following statements: (x 1, ) (x,y ) = (x 1, ) (, 3) 1 (x,y ) (0, 0) =(x 1 3, 1 3 ) (x,y ) (0, 0) =(x 1 3 + 1 3 x, 1 3 y ) (0, 0), (x 1, ) r (x,y ) = ((, 3) (x 1, ) 1 (x,y )) (, 3) =( 3x 1 + 3x, 3y ) (, 3),
96 Yong Chan Kim and Jin Won Park (x 1, ) l (x,y ) = ((x,y ) (x 1, ) 1 (, 3)) (, 3) =(x x 1y + y, 3y ) (, 3). It is not commutative because (1, 4 3 )=(3, ) ( 1, ) ( 1, ) (3, ) = ( 1 3, 4 3 ). Furthermore, we have (x, y) =(x, y) =(x, y) from: (x, y) =(x, y) r (0, 1) = ( 3x y, 3 y ), (x, y) =(x, y) l (0, 1) = ( x, 3 y y ). Other cases are proved as a similar method in Example.3(1). 3 Pseudo MV-algebras induced by functions Theorem 3.1 Let (L, ) be a lattice-group with identity e and P a positive cone. Let ψ : M L be a bijective function and define an operation ψ : M M M as x ψ y = ψ 1 (ψ(x) ψ(y)). (1) (M, ψ ) is a po-group with identity e ψ = ψ 1 (e), x 1 = ψ 1 (ψ(x) 1 ) and a positive cone ψ 1 (P ). () (M 1 [c, e ψ ], ψ, rψ, lψ,c,e ψ ) is a pseudo MV-algebra where c is the least element, e ψ is the greatest element and x ψ y =(x ψ y) c, x rψ y =(x 1 ψ y) e ψ, x lψ y =(y ψ x 1 ) e ψ. (3) (M + [e ψ,d], ψ, rψ, lψ,e ψ,d) is a pseudo MV-algebra where e ψ is the least element and d is the greatest element and x ψ y =(x ψ y ψ d 1 ) e ψ, x rψ y =(x 1 ψ y ψ d) d, x lψ y =(y ψ b ψ x 1 ) d. Proof. (1) We easily proved that (M, ψ ) is a group. Since e ψ = x x 1 = ψ 1 (ψ(x) ψ(x 1 )) implies e = ψ(x) ψ(x 1 ), we have ψ(x) 1 = ψ(x 1 ). So, ψ 1 (P 1 )=ψ 1 (P ) 1. Thus ψ 1 (P ) ψ 1 (P ) 1 = ψ 1 (P P 1 )={e ψ }. If x ψ y ψ 1 (P ) ψ ψ 1 (P ), then ψ(x) ψ(y) P, so, x ψ y ψ 1 (P ). Thus, ψ 1 (P ) ψ ψ 1 (P ) ψ 1 (P ). If a 1 ψ x ψ a a 1 ψ ψ 1 (P ) ψ a
Pseudo MV-algebras induced by functions 97 for x ψ 1 (P ), then a 1 ψ x ψ a = ψ 1 (ψ(a) 1 ψ(x) ψ(a)) ψ 1 (P ) because ψ(a) 1 ψ(x) ψ(a) ψ(a) 1 P ψ(a) =P.Ifx ψ 1 (P ), then ψ(x) P. Since ψ(a) ψ(x) ψ(a) 1 P, a ψ x ψ a 1 ψ 1 (P ). So, x a 1 ψ ψ 1 (P ) ψ a. Hence a 1 ψ ψ 1 (P ) ψ a = ψ 1 (P ). Thus ψ 1 (P ) is a positive cone. We can define a totally order ψ by ψ 1 (P ). By Lemma.1, (M, ψ, ψ ) is a lattice-group. () and (3) are similarly proved as in Theorem.. Example 3. Let L = {x R x>1} and M = {x R x>0} be two sets and an operation : L L L as x y = xy x y + with a positive cone P = {x L x }. Define ψ : M N as ψ(x) =x + 1. We obtain ψ : M M M as Then (M, ψ ) is a group with e ψ =1, x ψ y = ψ 1 (ψ(x) ψ(y)) = xy. x 1 = ψ 1 (ψ(x) 1 )= 1 x. We have a positive cone ψ 1 (P )={x M x 1}. We obtain x ψ y iff x 1 ψ y = y x 1. Then (M, ψ, ψ ) is a lattice-group. (1) (M =[ 1, 1], ψ, ψ, 1, 1) is an MV-algebra. Define x ψ y =(x ψ y) 1 = xy 1 x ψ y =(x 1 ψ y) 1= y x 1. We obtain x ψ (x ψ y)=x y and (x ψ y) (y ψ x) = 1. Moreover, 1 (x ψ ) ψ 1 = x. () (M + =[1, 3], ψ, ψ, 1, 3) is an MV-algebra. Define x ψ y =(x ψ 3 1 ψ y) 1= 1 (xy) 1, 3 x ψ y =(3 ψ x 1 ψ y) 3= 3y x 3. If x y, then x ψ (x ψ y)=x ψ 3=x. Ifx>y, then x ψ (x ψ y)=x ψ 3y = x. Hence x x ψ (x ψ y)=x y. Furthermore, (x ψ y) (y ψ x)=3. Moreover, (x ψ 1) ψ 1=x.
98 Yong Chan Kim and Jin Won Park Example 3.3 Let L = {(x, y) R y>0} and M = {(x, y) R x> 0} be two partially ordered sets and an operation : L L L as follows: (x 1, ) (x,y )=(x 1 + x, y ). with a positive cone P = {(x, y) R y =1,x 0, or y > 1} Define ψ : M N as ψ(x, y) =(y, x) and ψ : M M M as (,b 1 ) ψ (a,b ) = ψ 1 (ψ(,b 1 ) ψ(,b 1 )) = ψ 1 ((b 1, ) (b,a )) = ψ 1 (b 1 + b, a )=( a,b 1 + b ). Then (M, ψ ) is a group with e ψ =(1, 0) and (a, b) 1 = ψ 1 (ψ(a, b) 1 )=ψ 1 ( b a, 1 a )=(1 a, b a ). We have a positive cone ψ 1 (P )={(a, b) R a =1,b 0, or a>1}. For (,b 1 ), (a,b ) M, we define (,b 1 ) ψ (a,b ) (,b 1 ) 1 ψ (a,b ) ψ 1 (P ), (a,b ) ψ (,b 1 ) 1 ψ 1 (P ) <a or = a,b 1 b. Then (M, ψ ψ ) is a lattice-group. (1) (M [( 1, 1), (1, 0)], ψ, r, l, ( 1, 1), (1, 0)) is a pseudo MV-algebra where ( 1, 1) is the least element and (1, 0) is the greatest element from the following statements: (,b 1 ) ψ (a,b ) = (,b 1 ) ψ (a,b ) ( 1, 1) =( b,b 1 + b ) ( 1, 1), (,b 1 ) r (a,b ) = ((,b 1 ) 1 ψ (a,b )) (1, 0) =( a, b 1+b ) (1, 0), (,b 1 ) l (a,b ) = ((a,b ) (,b 1 ) 1 ) (1, 0) =( a,b b 1a ) (1, 0). () (M + [(1, 0), (, 3)],, r, l, (1, 0), (, 3)) is a pseudo MV-algebra where (1, 0) is the least element and (, 3) is the greatest element from the following statements: (,b 1 ) ψ (a,b ) = ( (,b 1 ) ψ (, 3) 1 ψ (a,b ) ) (1, 0) = ( (,b 1 3a 1) ψ (a,b ) ) (1, 0) =( 1a 1a,b 1 3a 1 + b ) (1, 0), (,b 1 ) r (a,b ) = ((, 3) ψ (,b 1 ) 1 ψ (a,b )) (, 3) =( a, 3+ b 1+b ) (, 3), (,b 1 ) l (a,b ) = ((a,b ) ψ (,b 1 ) 1 ψ (, 3)) (, 3) =( a,b + 3a a b 1 ) (, 3).
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