Pseudo MV-Algebras Induced by Functions
|
|
- Sharon Richards
- 5 years ago
- Views:
Transcription
1 International Mathematical Forum, 4, 009, no., Pseudo MV-Algebras Induced by Functions Yong Chan Kim Department of Mathematics, Kangnung National University, Gangneung, Gangwondo 10-70, Korea Jin Won Park Department of Mathematics Education, Cheju National University, Jeju, Jejudo , Korea 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.
2 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.
3 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.
4 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.
5 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).
6 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 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.
7 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 ) = ( , 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 x, 1 3 y ) (0, 0), (x 1, ) r (x,y ) = ((, 3) (x 1, ) 1 (x,y )) (, 3) =( 3x 1 + 3x, 3y ) (, 3),
8 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
9 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.
10 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).
11 Pseudo MV-algebras induced by functions 99 References [1] G. Birkhoff,Lattice Theory, Amer. Math. Soc. Colloq. Publ., vol 5, New York, [] C.C. Chang, Algebraic analysis of many valued logics, Trans. of A.M.S.,88()(1958), [3] A. Di Nola, P. Flondor, L. Leustean, MV-modules, J. of Algebra, 67(003), [4] A. Dvurecenskij, Pseudo MV-algebras are intervals in l-groups, J. Australian Math. Soc., 7(00), [5] A. Dvurecenskij, On pseudo MV-algebras, Soft Computing,5(001), [6] G. Georgescu, A. lorgulescu, Pseudo MV-algebras, Multiple-Valued Logics, 6(001), [7] G. Georgescu, A. Popescu, Non-commutative fuzzy structures and pairs of weak negations,fuzzy Sets and Systems, 143(004), [8] N. Galatos, C. Tsinakis, Generalized MV-alebras,J. of Algebra, 83(005), [9] P. Flonder, M. Sularia, On a class of residuated semilattice monoids,fuzzy Sets and Systems, 138(003), [10] U. Höhle, Many valued topology and its applications, Kluwer Academic Publisher, Boston, 001. [11] U. Höhle, E. P. Klement, Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publisher, Boston, [1] E. Turunen, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., Received: July, 008
Join Preserving Maps and Various Concepts
Int. J. Contemp. Math. Sciences, Vol. 5, 010, no. 5, 43-51 Join Preserving Maps and Various Concepts Yong Chan Kim Department of Mathematics, Gangneung-Wonju University Gangneung, Gangwondo 10-70, Korea
More informationDUAL BCK-ALGEBRA AND MV-ALGEBRA. Kyung Ho Kim and Yong Ho Yon. Received March 23, 2007
Scientiae Mathematicae Japonicae Online, e-2007, 393 399 393 DUAL BCK-ALGEBRA AND MV-ALGEBRA Kyung Ho Kim and Yong Ho Yon Received March 23, 2007 Abstract. The aim of this paper is to study the properties
More informationAndrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS
DEMONSTRATIO MATHEMATICA Vol. XLIII No 3 2010 Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS Abstract. The class of bipartite pseudo-bl algebras (denoted by BP) and the
More informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
More informationThe Properties of Various Concepts
International Journal of Fuzzy Logic and Intelligent Systems, vol. 10, no. 3, September 2010, pp.247-252 DOI : 10.5391/IJFIS.2010.10.3.247 The Properties of Various Concepts Yong Chan Kim and Jung Mi Ko
More informationThe Blok-Ferreirim theorem for normal GBL-algebras and its application
The Blok-Ferreirim theorem for normal GBL-algebras and its application Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics
More informationON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 28 (2008 ) 63 75 ON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS Grzegorz Dymek Institute of Mathematics and Physics University of Podlasie 3 Maja 54,
More informationMODAL, NECESSITY, SUFFICIENCY AND CO-SUFFICIENCY OPERATORS. Yong Chan Kim
Korean J. Math. 20 2012), No. 3, pp. 293 305 MODAL, NECESSITY, SUFFICIENCY AND CO-SUFFICIENCY OPERATORS Yong Chan Kim Abstract. We investigate the properties of modal, necessity, sufficiency and co-sufficiency
More informationFleas and fuzzy logic a survey
Fleas and fuzzy logic a survey Petr Hájek Institute of Computer Science AS CR Prague hajek@cs.cas.cz Dedicated to Professor Gert H. Müller on the occasion of his 80 th birthday Keywords: mathematical fuzzy
More informationEmbedding theorems for normal divisible residuated lattices
Embedding theorems for normal divisible residuated lattices Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics and Computer
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 58 (2009) 248 256 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Some
More informationSome Pre-filters in EQ-Algebras
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 2 (December 2017), pp. 1057-1071 Applications and Applied Mathematics: An International Journal (AAM) Some Pre-filters
More informationCLOSURE OPERATORS ON COMPLETE ALMOST DISTRIBUTIVE LATTICES-III
Bulletin of the Section of Logic Volume 44:1/2 (2015), pp. 81 93 G. C. Rao, Venugopalam Undurthi CLOSURE OPERATORS ON COMPLETE ALMOST DISTRIBUTIVE LATTICES-III Abstract In this paper, we prove that the
More informationPseudo-BCK algebras as partial algebras
Pseudo-BCK algebras as partial algebras Thomas Vetterlein Institute for Medical Expert and Knowledge-Based Systems Medical University of Vienna Spitalgasse 23, 1090 Wien, Austria Thomas.Vetterlein@meduniwien.ac.at
More informationMathematica Slovaca. Ján Jakubík On the α-completeness of pseudo MV-algebras. Terms of use: Persistent URL:
Mathematica Slovaca Ján Jakubík On the α-completeness of pseudo MV-algebras Mathematica Slovaca, Vol. 52 (2002), No. 5, 511--516 Persistent URL: http://dml.cz/dmlcz/130365 Terms of use: Mathematical Institute
More informationWEAK EFFECT ALGEBRAS
WEAK EFFECT ALGEBRAS THOMAS VETTERLEIN Abstract. Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is
More informationObstinate filters in residuated lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103) No. 4, 2012, 413 422 Obstinate filters in residuated lattices by Arsham Borumand Saeid and Manijeh Pourkhatoun Abstract In this paper we introduce the
More informationPartial, Total, and Lattice Orders in Group Theory
Partial, Total, and Lattice Orders in Group Theory Hayden Harper Department of Mathematics and Computer Science University of Puget Sound April 23, 2016 Copyright c 2016 Hayden Harper. Permission is granted
More informationFuzzy relation equations with dual composition
Fuzzy relation equations with dual composition Lenka Nosková University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1 Czech Republic Lenka.Noskova@osu.cz
More informationON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 35 (2015) 21 31 doi:10.7151/dmgaa.1227 ON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS Grzegorz Dymek Institute of Mathematics and Computer Science
More informationITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 631
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 2017 (631 642) 631 n-fold (POSITIVE) IMPLICATIVE FILTERS OF HOOPS Chengfang Luo Xiaolong Xin School of Mathematics Northwest University Xi an 710127
More informationSome remarks on hyper MV -algebras
Journal of Intelligent & Fuzzy Systems 27 (2014) 2997 3005 DOI:10.3233/IFS-141258 IOS Press 2997 Some remarks on hyper MV -algebras R.A. Borzooei a, W.A. Dudek b,, A. Radfar c and O. Zahiri a a Department
More informationSEMIRINGS SATISFYING PROPERTIES OF DISTRIBUTIVE TYPE
proceedings of the american mathematical society Volume 82, Number 3, July 1981 SEMIRINGS SATISFYING PROPERTIES OF DISTRIBUTIVE TYPE ARIF KAYA AND M. SATYANARAYANA Abstract. Any distributive lattice admits
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
More informationFuzzy Logics and Substructural Logics without Exchange
Fuzzy Logics and Substructural Logics without Exchange Mayuka F. KAWAGUCHI Division of Computer Science Hokkaido University Sapporo 060-0814, JAPAN mayuka@main.ist.hokudai.ac.jp Osamu WATARI Hokkaido Automotive
More informationFUZZY BCK-FILTERS INDUCED BY FUZZY SETS
Scientiae Mathematicae Japonicae Online, e-2005, 99 103 99 FUZZY BCK-FILTERS INDUCED BY FUZZY SETS YOUNG BAE JUN AND SEOK ZUN SONG Received January 23, 2005 Abstract. We give the definition of fuzzy BCK-filter
More information2 Basic Results on Subtraction Algebra
International Mathematical Forum, 2, 2007, no. 59, 2919-2926 Vague Ideals of Subtraction Algebra Young Bae Jun Department of Mathematics Education (and RINS) Gyeongsang National University, Chinju 660-701,
More informationStrong Tensor Non-commutative Residuated Lattices
Strong Tensor Non-commutative Residuated Lattices Hongxing Liu Abstract In this paper, we study the properties of tensor operators on non-commutative residuated lattices. We give some equivalent conditions
More informationarxiv: v1 [math-ph] 23 Jul 2010
EXTENSIONS OF WITNESS MAPPINGS GEJZA JENČA arxiv:1007.4081v1 [math-ph] 23 Jul 2010 Abstract. We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose
More informationMonadic GMV -algebras
Department of Algebra and Geometry Faculty of Sciences Palacký University of Olomouc Czech Republic TANCL 07, Oxford 2007 monadic structures = algebras with quantifiers = algebraic models for one-variable
More information``Residuated Structures and Many-valued Logics''
1 Conference on ``Residuated Structures and Many-valued Logics'' Patras, 2-5 June 2004 Abstracts of Invited Speakers 2 FRAMES AND MV-ALGEBRAS L. P. Belluce and A. Di Nola Department of Mathematics British
More informationSTRONGLY EXTENSIONAL HOMOMORPHISM OF IMPLICATIVE SEMIGROUPS WITH APARTNESS
SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (26), No.2, (2017), 155 162 DOI: 10.5644/SJM.13.2.03 STRONGLY EXTENSIONAL HOMOMORPHISM OF IMPLICATIVE SEMIGROUPS WITH APARTNESS DANIEL ABRAHAM ROMANO Abstract. The
More informationLecture 8: Equivalence Relations
Lecture 8: Equivalence Relations 1 Equivalence Relations Next interesting relation we will study is equivalence relation. Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation on
More informationA Survey of Generalized Basic Logic Algebras
A Survey of Generalized Basic Logic Algebras Nikolaos Galatos and Peter Jipsen abstract. Petr Hájek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval,
More informationCross Connection of Boolean Lattice
International Journal of Algebra, Vol. 11, 2017, no. 4, 171-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7419 Cross Connection of Boolean Lattice P. G. Romeo P. R. Sreejamol Dept.
More informationOn the lattice of congruence filters of a residuated lattice
Annals of University of Craiova, Math. Comp. Sci. Ser. Volume 33, 2006, Pages 174 188 ISSN: 1223-6934 On the lattice of congruence filters of a residuated lattice Raluca Creţan and Antoaneta Jeflea Abstract.
More informationOn injective constructions of S-semigroups. Jan Paseka Masaryk University
On injective constructions of S-semigroups Jan Paseka Masaryk University Joint work with Xia Zhang South China Normal University BLAST 2018 University of Denver, Denver, USA Jan Paseka (MU) 10. 8. 2018
More informationIDEALS AND THEIR FUZZIFICATIONS IN IMPLICATIVE SEMIGROUPS
International Journal of Pure and Applied Mathematics Volume 104 No. 4 2015, 543-549 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v104i4.6
More informationFrom Residuated Lattices to Boolean Algebras with Operators
From Residuated Lattices to Boolean Algebras with Operators Peter Jipsen School of Computational Sciences and Center of Excellence in Computation, Algebra and Topology (CECAT) Chapman University October
More informationPrime and Irreducible Ideals in Subtraction Algebras
International Mathematical Forum, 3, 2008, no. 10, 457-462 Prime and Irreducible Ideals in Subtraction Algebras Young Bae Jun Department of Mathematics Education Gyeongsang National University, Chinju
More informationApproximating models based on fuzzy transforms
Approximating models based on fuzzy transforms Irina Perfilieva University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic e-mail:irina.perfilieva@osu.cz
More informationAn embedding of ChuCors in L-ChuCors
Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010 27 30 June 2010. An embedding of ChuCors in L-ChuCors Ondrej Krídlo 1,
More informationThe structure of many-valued relations
The structure of many-valued relations M.Emilia Della Stella 1 dellastella@science.unitn.it Cosimo Guido 2 cosimo.guido@unisalento.it 1 Department of Mathematics-University of Trento 2 Department of Mathematics-University
More informationProbabilistic averaging in bounded commutative residuated l-monoids
Discrete Mathematics 306 (2006) 1317 1326 www.elsevier.com/locate/disc Probabilistic averaging in bounded commutative residuated l-monoids Anatolij Dvurečenskij a, Jiří Rachůnek b a Mathematical Institute,
More informationA NOTE ON FOUR TYPES OF REGULAR RELATIONS. H. S. Song
Korean J. Math. 20 (2012), No. 2, pp. 177 184 A NOTE ON FOUR TYPES OF REGULAR RELATIONS H. S. Song Abstract. In this paper, we study the four different types of relations, P(X, T ), R(X, T ), L(X, T ),
More informationFuzzy Function: Theoretical and Practical Point of View
EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France Fuzzy Function: Theoretical and Practical Point of View Irina Perfilieva, University of Ostrava, Inst. for Research and Applications of Fuzzy Modeling,
More informationMODAL OPERATORS ON COMMUTATIVE RESIDUATED LATTICES. 1. Introduction
ao DOI: 10.2478/s12175-010-0055-1 Math. Slovaca 61 (2011), No. 1, 1 14 MODAL OPERATORS ON COMMUTATIVE RESIDUATED LATTICES M. Kondo (Communicated by Jiří Rachůnek ) ABSTRACT. We prove some fundamental properties
More informationSome properties of residuated lattices
Some properties of residuated lattices Radim Bělohlávek, Ostrava Abstract We investigate some (universal algebraic) properties of residuated lattices algebras which play the role of structures of truth
More informationContents. Introduction
Contents Introduction iii Chapter 1. Residuated lattices 1 1. Definitions and preliminaries 1 2. Boolean center of a residuated lattice 10 3. The lattice of deductive systems of a residuated lattice 14
More informationEQ-algebras: primary concepts and properties
UNIVERSITY OF OSTRAVA Institute for Research and Applications of Fuzzy Modeling EQ-algebras: primary concepts and properties Vilém Novák Research report No. 101 Submitted/to appear: Int. Joint, Czech Republic-Japan
More informationOn the filter theory of residuated lattices
On the filter theory of residuated lattices Jiří Rachůnek and Dana Šalounová Palacký University in Olomouc VŠB Technical University of Ostrava Czech Republic Orange, August 5, 2013 J. Rachůnek, D. Šalounová
More informationOn Regularity of Incline Matrices
International Journal of Algebra, Vol. 5, 2011, no. 19, 909-924 On Regularity of Incline Matrices A. R. Meenakshi and P. Shakila Banu Department of Mathematics Karpagam University Coimbatore-641 021, India
More informationA Generalization of Generalized Triangular Fuzzy Sets
International Journal of Mathematical Analysis Vol, 207, no 9, 433-443 HIKARI Ltd, wwwm-hikaricom https://doiorg/02988/ijma2077350 A Generalization of Generalized Triangular Fuzzy Sets Chang Il Kim Department
More informationCONSTRUCTIVE GELFAND DUALITY FOR C*-ALGEBRAS
CONSTRUCTIVE GELFAND DUALITY FOR C*-ALGEBRAS THIERRY COQUAND COMPUTING SCIENCE DEPARTMENT AT GÖTEBORG UNIVERSITY AND BAS SPITTERS DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, EINDHOVEN UNIVERSITY OF
More informationPublications since 2000
Publications since 2000 Wolfgang Rump 1. Two-Point Differentiation for General Orders, Journal of Pure and Applied Algebra 153 (2000), 171-190. 2. Representation theory of two-dimensional Brauer graph
More informationSummary of the PhD Thesis MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Lukasiewicz logic Serafina Lapenta
Summary of the PhD Thesis MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Lukasiewicz logic Serafina Lapenta The framework. In 1956, G. Birkhoff G. and R.S. Pierce [1] conjectured
More informationPartial Metrics and Quantale-valued Sets. by Michael Bukatin, Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh
Michael Bukatin presents: Partial Metrics and Quantale-valued Sets by Michael Bukatin, Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh http://www.cs.brandeis.edu/ bukatin/distances and equalities.html
More informationGENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS. Chun Gil Park
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 3 (003), 183 193 GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS Chun Gil Park (Received March
More informationSoft set theoretical approach to residuated lattices. 1. Introduction. Young Bae Jun and Xiaohong Zhang
Quasigroups and Related Systems 24 2016, 231 246 Soft set theoretical approach to residuated lattices Young Bae Jun and Xiaohong Zhang Abstract. Molodtsov's soft set theory is applied to residuated lattices.
More informationON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES. 1. Introduction
Math. Appl. 5 (2016, 39 53 DOI: 10.13164/ma.2016.04 ON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES RADEK ŠLESINGER Abstract. If the standard concepts of partial-order relation
More informationWhen does a semiring become a residuated lattice?
When does a semiring become a residuated lattice? Ivan Chajda and Helmut Länger arxiv:1809.07646v1 [math.ra] 20 Sep 2018 Abstract It is an easy observation that every residuated lattice is in fact a semiring
More informationIntuitionistic L-Fuzzy Rings. By K. Meena & K. V. Thomas Bharata Mata College, Thrikkakara
Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 14 Version 1.0 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More informationSome Generalizations of Caristi s Fixed Point Theorem with Applications
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 12, 557-564 Some Generalizations of Caristi s Fixed Point Theorem with Applications Seong-Hoon Cho Department of Mathematics Hanseo University, Seosan
More informationIDEMPOTENT ELEMENTS OF THE ENDOMORPHISM SEMIRING OF A FINITE CHAIN
Доклади на Българската академия на науките Comptes rendus de l Académie bulgare des Sciences Tome 66, No 5, 2013 MATHEMATIQUES Algèbre IDEMPOTENT ELEMENTS OF THE ENDOMORPHISM SEMIRING OF A FINITE CHAIN
More informationDerivations on Trellises
Journal of Applied & Computational Mathematics Journal of Applied & Computational Mathematics Ebadi and Sattari, J Appl Computat Math 2017, 7:1 DOI: 104172/2168-96791000383 Research Article Open Access
More informationL fuzzy ideals in Γ semiring. M. Murali Krishna Rao, B. Vekateswarlu
Annals of Fuzzy Mathematics and Informatics Volume 10, No. 1, (July 2015), pp. 1 16 ISSN: 2093 9310 (print version) ISSN: 2287 6235 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com
More informationON STRUCTURE OF KS-SEMIGROUPS
International Mathematical Forum, 1, 2006, no. 2, 67-76 ON STRUCTURE OF KS-SEMIGROUPS Kyung Ho Kim Department of Mathematics Chungju National University Chungju 380-702, Korea ghkim@chungju.ac.kr Abstract
More informationTHE notion of fuzzy groups defined by A. Rosenfeld[13]
I-Vague Groups Zelalem Teshome Wale Abstract The notions of I-vague groups with membership and non-membership functions taking values in an involutary dually residuated lattice ordered semigroup are introduced
More information212 J. MENG AND Y. B. JUN An ideal I is said to be prime if (v) x ^ y 2 I implies x 2 I or y 2 I. The set of all ideals of X is denoted by I(X); the s
Scientiae Mathematicae Vol.1, No. 2(1998), 211{215 211 THE SPECTRAL SPACE OF MV{ALGEBRAS IS A STONE SPACE Jie Meng and Young Bae Jun y Received July 20, 1995 Abstract. Let X be an MV-algebra and let Spec(X)
More informationPrimitive Ideals of Semigroup Graded Rings
Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics Department 2004 Primitive Ideals of Semigroup Graded Rings Hema Gopalakrishnan Sacred Heart University, gopalakrishnanh@sacredheart.edu
More informationPhysical justification for using the tensor product to describe two quantum systems as one joint system
Physical justification for using the tensor product to describe two quantum systems as one joint system Diederik Aerts and Ingrid Daubechies Theoretical Physics Brussels Free University Pleinlaan 2, 1050
More informationTense Operators on Basic Algebras
Int J Theor Phys (2011) 50:3737 3749 DOI 10.1007/s10773-011-0748-4 Tense Operators on Basic Algebras M. Botur I. Chajda R. Halaš M. Kolařík Received: 10 November 2010 / Accepted: 2 March 2011 / Published
More informationPARAMETRIC OPERATIONS FOR TWO FUZZY NUMBERS
Commun. Korean Math. Soc. 8 (03), No. 3, pp. 635 64 http://dx.doi.org/0.434/ckms.03.8.3.635 PARAMETRIC OPERATIONS FOR TWO FUZZY NUMBERS Jisoo Byun and Yong Sik Yun Abstract. There are many results on the
More informationMV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS. (Received August 27, 1997)
123 (1998) MATHEMATICA BOHEMICA No. 4, 437 441 MV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS Jiří Rachůnek, Olomouc (Received August 27, 1997) Abstract. In the paper it is
More informationarxiv: v1 [math.ra] 1 Apr 2015
BLOCKS OF HOMOGENEOUS EFFECT ALGEBRAS GEJZA JENČA arxiv:1504.00354v1 [math.ra] 1 Apr 2015 Abstract. Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalize some
More informationSung-Wook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,
More informationOn Homomorphism and Algebra of Functions on BE-algebras
On Homomorphism and Algebra of Functions on BE-algebras Kulajit Pathak 1, Biman Ch. Chetia 2 1. Assistant Professor, Department of Mathematics, B.H. College, Howly, Assam, India, 781316. 2. Principal,
More informationRepresentation of States on MV-algebras by Probabilities on R-generated Boolean Algebras
Representation of States on MV-algebras by Probabilities on R-generated Boolean Algebras Brunella Gerla 1 Tomáš Kroupa 2,3 1. Department of Informatics and Communication, University of Insubria, Via Mazzini
More informationClosure operators on sets and algebraic lattices
Closure operators on sets and algebraic lattices Sergiu Rudeanu University of Bucharest Romania Closure operators are abundant in mathematics; here are a few examples. Given an algebraic structure, such
More informationAN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC
Iranian Journal of Fuzzy Systems Vol. 9, No. 6, (2012) pp. 31-41 31 AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC E. ESLAMI Abstract. In this paper we extend the notion of degrees of membership
More informationComputing in lattice ordered groups and related structures
Computing in lattice ordered groups and related structures Peter Jipsen Chapman University, Orange, California, USA June 16, 2009 Peter Jipsen (Chapman) Computing in l-groups June 16, 2009 1 / 23 Outline
More informationON GENERAL BEST PROXIMITY PAIRS AND EQUILIBRIUM PAIRS IN FREE GENERALIZED GAMES 1. Won Kyu Kim* 1. Introduction
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No.1, March 2006 ON GENERAL BEST PROXIMITY PAIRS AND EQUILIBRIUM PAIRS IN FREE GENERALIZED GAMES 1 Won Kyu Kim* Abstract. In this paper, using
More informationRESIDUATION SUBREDUCTS OF POCRIGS
Bulletin of the Section of Logic Volume 39:1/2 (2010), pp. 11 16 Jānis Cīrulis RESIDUATION SUBREDUCTS OF POCRIGS Abstract A pocrig (A,,, 1) is a partially ordered commutative residuated integral groupoid.
More informationYONGDO LIM. 1. Introduction
THE INVERSE MEAN PROBLEM OF GEOMETRIC AND CONTRAHARMONIC MEANS YONGDO LIM Abstract. In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed
More informationRESTRICTED FLEXIBLE ALGEBRAS
J. Korean Math. Soc. 25(1988), No.1, 99. 109-113 RESTRICTED FLEXIBLE ALGEBRAS YOUNGSO Ko* AND HyO CHUL MYUNG+ 1. Introduction An algebra A with multiplication denoted by xy over a field F is called flexible
More informationFrom Semirings to Residuated Kleene Lattices
Peter Jipsen From Semirings to Residuated Kleene Lattices Abstract. We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-. An investigation
More informationThe Square of Opposition in Orthomodular Logic
The Square of Opposition in Orthomodular Logic H. Freytes, C. de Ronde and G. Domenech Abstract. In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative,
More informationThe logic of perfect MV-algebras
The logic of perfect MV-algebras L. P. Belluce Department of Mathematics, British Columbia University, Vancouver, B.C. Canada. belluce@math.ubc.ca A. Di Nola DMI University of Salerno, Salerno, Italy adinola@unisa.it
More informationLaw of total probability and Bayes theorem in Riesz spaces
Law of total probability and Bayes theorem in Riesz spaces Liang Hong Abstract. This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationAPPROXIMATIONS IN H v -MODULES. B. Davvaz 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 6, No. 4, pp. 499-505, December 2002 This paper is available online at http://www.math.nthu.edu.tw/tjm/ APPROXIMATIONS IN H v -MODULES B. Davvaz Abstract. In this
More informationOn Symmetric Bi-Multipliers of Lattice Implication Algebras
International Mathematical Forum, Vol. 13, 2018, no. 7, 343-350 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2018.8423 On Symmetric Bi-Multipliers of Lattice Implication Algebras Kyung Ho
More informationDOUGLAS J. DAILEY AND THOMAS MARLEY
A CHANGE OF RINGS RESULT FOR MATLIS REFLEXIVITY DOUGLAS J. DAILEY AND THOMAS MARLEY Abstract. Let R be a commutative Noetherian ring and E the minimal injective cogenerator of the category of R-modules.
More informationJORDAN HOMOMORPHISMS AND DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS
JORDAN HOMOMORPHISMS AND DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS A. M. SINCLAIR 1. Introduction. One may construct a Jordan homomorphism from one (associative) ring into another ring by taking the sum
More informationEQ-ALGEBRAS WITH PSEUDO PRE-VALUATIONS. Yongwei Yang 1. Xiaolong Xin
italian journal of pure and applied mathematics n. 37 2017 (29 48) 29 EQ-ALGEBRAS WITH PSEUDO PRE-VALUATIONS Yongwei Yang 1 School of Mathematics and Statistics Anyang Normal University Anyang 455000 China
More informationPrime and Semiprime Bi-ideals in Ordered Semigroups
International Journal of Algebra, Vol. 7, 2013, no. 17, 839-845 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.310105 Prime and Semiprime Bi-ideals in Ordered Semigroups R. Saritha Department
More informationSubalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant fuzzy set theory
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 417-430 ISSN 1307-5543 www.ejpam.com Published by New York Business Global Subalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant
More informationA TALE OF TWO CONFORMALLY INVARIANT METRICS
A TALE OF TWO CONFORMALLY INVARIANT METRICS H. S. BEAR AND WAYNE SMITH Abstract. The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationAN INVERSE MATRIX OF AN UPPER TRIANGULAR MATRIX CAN BE LOWER TRIANGULAR
Discussiones Mathematicae General Algebra and Applications 22 (2002 ) 161 166 AN INVERSE MATRIX OF AN UPPER TRIANGULAR MATRIX CAN BE LOWER TRIANGULAR Waldemar Ho lubowski Institute of Mathematics Silesian
More information