Advanced
Fuzzy relation equations in pseudo BL-algebras
Fuzzy relation equations in pseudo BL-algebras
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Sep, 13(3): 208-214
Copyright ©2013, Korean Institute of Intelligent Systems
This is an Open Access article distributedunder the terms of the CreativeCommons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercialuse, distribution, and reproductionin any medium, provided the originalwork is properly cited.
  • Received : December 18, 2012
  • Accepted : September 17, 2013
  • Published : September 25, 2013
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Yong Chan Kim

Abstract
Bandler and Kohout investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.
Keywords
1. Introduction
Sanchez [1] introduced the theory of fuzzy relation equations with various types of compositions: max-min, min-max, and min- α . Fuzzy relation equations with new types of compositions (continuous t-norm and residuated lattice) have been developed [2 - 5] . In particular, Bandler and Kohout [6] investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In contrast, noncommutative structures play an important role in metric spaces and algebraic structures (groups, rings, quantales, and pseudo BL-algebras) [7 - 15] . Georgescu and Iorgulescu [12] introduced pseudo MV-algebras as the generalization of MV-algebras. Georgescu and Leustean [11] introduced generalized residuated lattice as a noncommutative structure.
In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions Ai R = Bi and Ai R = Bi in pseudo BL-algebras.
2. Preliminaries
Definition 2.1. [11] A structure ( L , ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is called apseudo BL-algebra if it satisfies the following conditions:
(A1) ( L , ˅, ˄, →, ⊤, ⊥) is bounded where ⊤ is the universal upper bound and ⊥ denotes the universal lower bound;
(A2) ( L , ⊙, ⊤) is a monoid;
(A3) it satisfies a residuation, i.e.,
  • a⊙b≤ciffa≤b→ciffb≤a⇒c.
(A4) a ˄ b = ( a b ) ⊙ a = a ⊙ ( a b ).
(A5) ( a b ) ˅ ( b a ) = ⊤ and ( a b ) ˅ ( b a ) = ⊤.
We denote a 0 = a →⊥ and a = a ⇒⊥.
A pseudo BL-chain is a linear pseudo BL-algebra, i.e., a pseudo BL-algebra such that its lattice order is total.
In this paper, we assume that ( L , ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is a pseudo BL-algebra.
Lemma 2.2. [11] For each x , y , z , xi , yi L , we have the following properties:
(1) If y z , ( x y ) ≤ ( x z ), x y x z , and z x y x for →∈ {→, ⇒}.
(2) x y x y x y .
(3) ( x y ) → z = x → ( y z ) and ( x y ) ⇒ z = y ⇒ ( x z ).
(4) x → ( y z )= y ⇒ ( x z ) and x ⇒ ( y z )= y → ( x z ).
(5) x ⊙ ( x y ) ≤ y and ( x y ) ⊙ x y .
(6) x ⊙ ( y z )=( x y ) ∨ ( x z ) and ( x y ) ⊙ z = ( x z ) ∨ ( y z ).
(7) x y = ⊤ iff x y iff x y = ⊤
3. Fuzzy Relation Equations in Pseudo BL-Algebras
Theorem 3.1. Let a =( a 1 , a 2 , ..., an ) ∈ Ln and b L . We define two equations with respect to an unknown x = ( x 1 , ..., ( xn ) ∈ Ln as
Lager Image
Lager Image
Then, (1) (I) is solvable iff it has the least solution y = ( y 1 , ..., yn ) ∈ Ln such that yj = b aj , j =1, ..., n .
(2) (II) is solvable iff it has the least solution x =(( x 1 , ..., ( xn ) ∈ Ln such that xj = aj b , j =1, ..., n .
(3) If (I) is solvable, then
Lager Image
(4) If (II) is solvable, then
Lager Image
Proof . (1) (⇒) Let x =(( x 1 , ..., ( xn ) be a solution of (I). Since
Lager Image
Moreover,
Lager Image
Thus, y = ( b a 1 , ..., b an ) is the least solution.
(⇐) It is trivial.
(3) Let x =( x 1 , ..., xn ) denote a solution of (I). Then, b =
Lager Image
(2) and (4) are similarly proved as (1) and (3), respectively.
Theorem 3.2. Let L denote a pseudo BL-chain in equations (I) and (II) of Theorem 3.1.
(1) If b ˂ ⊤ and
Lager Image
with B = { ajk |1 ≤ k m , b = ( ajk )*}, then
Lager Image
is a maximal solution of (II). Moreover, if x is a solution of (II), there exists k ∈{ jk |1 ≤ k m } such that
xjk = 0, j = k , xj aj b , j k
where there exists x jk X such that x ≤ x jk .
(2) If b ˂⊤ and
Lager Image
with B = { ajk |1 ≤ k m , b = ( ajk ) 0 }, then
Lager Image
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists k ∈ { jk |1 ≤ k m } such that
  • xjk= 0,j=k,xj≥b⊙aj,j≠k
where there exists x jk X such that x ≤ x jk .
Proof . (1) (⇒)
Lager Image
is a solution of (II) because
Lager Image
Let x ≥ x jk be a solution of (II). Then,
Lager Image
with xjk ajk b and
Lager Image
Since b ˂ 1,
Lager Image
Since L is linear, ajk > xjk . Since
Lager Image
we have
  • xjk=ajk∧xjk=ajk⊙ (ajk⇒xjk) =ajk⊙b=ajk⊙ (ajk⇒⊥)= ⊥.
Thus, x = x jk .
Lager Image
is a maximal solution of (II).
Let x =( x 1 , ..., xn ) be a solution of (II). Since
Lager Image
by the linearity of L , there exists a family K = { jk | ajk B , ajk ⇒⊥ = b , 1 ≤ k m } such that
Lager Image
, because by linearity of L , ajk B , ( aj ) > b implies that
Lager Image
For k K , since ak ⇒⊥ = ak xk = b ≠ ⊤ and L is linear, ak > xk and ak b = ak ⊙ ( ak xk )= ak ⊙ ( ak ⇒ ⊥) = ⊥ ak xk = xk . Then,
Lager Image
(⇐) It is trivial.
(2) It is similarly proved as (1).
Example 3.3. Let K = {( x , y ) ∈ R 2 | x > 0} denote a set, and we define an operation ⊗ : K × K K as follows:
  • (x1,y1) ⊗ (x2,y2)=(x1x2,x1y2+y1).
Then, ( K , ⊗) is a group with e = (1, 0),
Lager Image
We have a positive cone P = {( a , b ) ∈ R 2 | a =1, b ≥ 0, or a > 1} because P P −1 = {(1, 0)}, P P P , ( a , b ) −1 P ⊙ ( a , b )= P , and P P −1 = K . For ( x 1 , y 1 ), ( x 2 , y 2 ) ∈ K , we define
( x 1 , y 1 ) ≤ ( x 2 , y 2 ) ⇔ ( x 1 , y 1 ) −1 ⊙ ( x 2 , y 2 ) ∈ P , ( x 2 , y 2 ) ⊙ ( x 1 , y 1 ) −1 P x 1 ˂ x 2 or x 1 = x 2 , y 1 y 2 .
Then, ( K , ≤⊗) is a lattice-group with totally order ≤. (ref. [1] )
The structure
Lager Image
is a Pseudo BL-chain where
Lager Image
is the least element and ⊤ = (1, 0) is the greatest element from the following statements:
Lager Image
Furthermore, we have ( x , y )=( x , y ) ∗◦ =( x , y ) ◦∗ from:
Lager Image
(1) An equation is defined as
Lager Image
Since
Lager Image
by Theorem 3.1(3), it is not solvable.
(2) An equation is defined as
Lager Image
Since
Lager Image
  • X= {x = ((x1,y1), (x2,y2), ⊥) or x = ((x1,y1), ⊥, (x3,y3)) | (x1,y1), (x2,y2), (x3,y3) ≥ ⊥}
is a solution set of (I).
M = {(⊤, ⊤, ⊥), (⊤, ⊥, ⊤)} is a maximal solution family of (I).
(3) An equation is defined as
Lager Image
Since
Lager Image
by Theorem 3.1(3), it is not solvable.
(4) An equation is defined as
Lager Image
Since
Lager Image
X = {x = (( x 1 , y 1 ), ( x 2 , y 2 ), ⊥) | ( x 1 , y 1 ), ( x 2 , y 2 ) ≥ ⊥} is a solution family of (II). (⊤, ⊤, ⊥) is a maximal solution of (II).
Definition 3.4. Let L denote a pseudo BL-chain. L satisfies the right conditional cancellation law if
  • ⊤ ˂a⊙x≤a⊙y⇒x≤y.
L satisfies the left conditional cancellation law if
  • ⊤ ˂x⊙a≤y⊙a⇒x≤y.
Theorem 3.5. Let L denote a pseudo BL-chain in two equations (I) and (II) of Theorem 3.1.
Then, (1) If L satisfies the right conditional cancellation law b ˂ ⊤ and
Lager Image
with B = { ajk |1 ≤ k m , b ˃ ( ajk ) * }, then
Lager Image
is a maximal solution family of (II). Moreover, if x is a solution of (II), there exists a family K = { jk | ajk B , ajk xjk = b , 1 ≤ k m } such that
  • xk=ak⊙b,k∈K,xj≥aj⊙b,j∉K
where there exists x jk X such that x ≤ x jk .
(2) If L satisfies the left conditional cancellation law b ˂ ⊤ and
Lager Image
with B = { ajk | 1 ≤ k m , b = ( ajk ) 0 }, then
Lager Image
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists k ∈ { jk | 1 ≤ k m } such that
  • xk=b⊙ak,j=k,xj≥b⊙aj,j≠k
where there exists x jk X such that x ≤ x jk .
Proof . (1)
Lager Image
is a solution of (II) because
Lager Image
Let x ≥ x jk denote a solution of (II). Then,
Lager Image
with xjk ajk b and
Lager Image
Since b ˂ 1,
Lager Image
Since L is linear, ajk ˃ xjk . Thus,
  • xjk=ajk∧xjk=ajk⊙ (ajk⇒xjk) =ajk⊙b.
Therefore, x = x jk .
Lager Image
is a maximal solution of (II).
Let x =( x 1 , ..., xn ) denote a solution of (II). Since
Lager Image
by the linearity of L , there exists a family K = { jk | ajk B , ajk xjk = b , 1 ≤ k m } such that
Lager Image
because ajk B , ( aj ) 0 b implies that
Lager Image
For k K , since ak xk = b ≠ ⊤ and L is linear, ak > xk and ak b = ak ⊙ ( ak xk )= ak xk = xk . For j K , since aj xj b , xj aj b . Hence,
  • xk=ak⊙b,k∈K,xj≥aj⊙b,j∉K
(⇐) It is trivial.
(2) It is similarly proved as (1).
Example 3.6. The structure
Lager Image
is defined as that in Example 3.3. Then, L satisfies the right conditional cancellation law because
  • ⊥ ˂ (a,b) ⊙ (x1,y1) ≤ (a,b) ⊙ (x2,y2)
  • (⇔)⊥ ˂ (ax1,ay1+b) ≤ (ax2,ay2+b)
  • (⇒)ax1=ax2,ay1+b≤ay1+b, orax1˂ax2
  • (⇒)x1=x2,y1≤y1, orx1˂x2
  • (⇒)(x1,y1)
  • ≤ (x2,y2).
Similarly, L satisfies the left conditional cancellation law.
(1) An equation is defined as
Lager Image
Since
Lager Image
is a maximal solution of (II) because
Lager Image
X = {x = (( x 1 , y 1 ), ( x 2 , y 2 ), ⊥ | ( x 1 , y 1 ), ( x 2 , y 2 ) ≥ ⊥} is a solution set of (II).
(2) An equation is defined as
Lager Image
Since
Lager Image
and
Lager Image
and
Lager Image
are maximal solutions of (II) because
Lager Image
is a solution set of (II).
(3) An equation is defined as
Lager Image
Since
Lager Image
  • X= {x = ((x1,y1), (x2,y2), ⊥) or x = (x1,y1), ⊥, (x3,y3)) | (x1,y1), (x2,y2), (x3,y3) ≥ ⊥}
is a solution set of (I).
Theorem 3.7. Let ai =( a i1 , a i2 , ..., ain ) ∈ Ln and bi L . We define two equations with respect to an unknown x = ( x 1 , ..., xn ) ∈ Ln as
Lager Image
Lager Image
Then, (1) (III) is solvable iff it has the least solution x = ( x 1 , ..., xn ) ∈ Ln such that
Lager Image
(2) (IV) is solvable iff it has the least solution x = ( x 1 , ..., xn ) ∈ Ln such that
Lager Image
(3) If (III) is solvable, then
Lager Image
(4) If (IV) is solvable, then
Lager Image
(5) If (III) (resp. (IV)) is solvable and x 1 , ..., x m is a solution of each ith equation, i =1, 2, ..., m , then
Lager Image
is a solution of (III) (resp. (IV)). Moreover, if each solution xi of the ith equation is maximal, any maximal solution x of (III) (resp. (IV)) is
Lager Image
Proof . (1) (⇒) Let y =( y 1 , ..., yn ) denote a solution of (III). Since
Lager Image
Moreover,
Lager Image
Then,
Lager Image
Substitute
Lager Image
Thus, ( x 1 , ..., xn ) is the least solution.
(⇐) It is trivial.
(3)
Lager Image
(2) and (4) are similarly proved as (1) and (3), respectively.
5) Let xi =( x i1 , ..., xin ) denote a solution of the ith equation in (III) and
Lager Image
Then,
Lager Image
Moreover,
Lager Image
Hence,
Lager Image
Therefore,
Lager Image
is a solution of (III).
Moreover, if xi is a maximal solution of the ith equation in (III), then
Lager Image
is a solution of (III). Let y = ( y 1 , ..., yn ) denote a solution of (III). Then, y ≤ x i for each i = 1, ... m . Then,
Lager Image
Hence, x is a maximal solution of (III).
Example 3.8. The structure
Lager Image
is defined as that in Example 3.3.
(1) An equation is defined as
Lager Image
Lager Image
is a solution set.
Lager Image
is a maximal solution set.
(2) An equation is defined as
Lager Image
is a solution set.
Lager Image
is a maximal solution set.
Lager Image
is a solution set of (1) and (2).
Lager Image
is a maximal solution set of (1) and (2).
(3) An equation is defined as
Lager Image
Lager Image
is a solution set.
Lager Image
is a maximal solution set.
(4) An equation is defined as
Lager Image
Lager Image
is a solution set.
Lager Image
is a maximal solution set.
Lager Image
is a solution set of (3) and (4).
Lager Image
is a maximal solution set of (3) and (4).
4. Conclusion
Bandler and Kohout [6] investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigated various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.
In the future, we will investigate various solutions of fuzzy relation equations with sup-compositions in pseudo BL-algebras and other algebraic structures.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the Research Institute of NaturalScience of Gangneung-Wonju National University.
References
Sanchez E. 1976 ”Resolution of composite fuzzy relation equations” Information and Control 30 (1) 38 - 48
Gottwald S. 1984 ”On the existence of solutions of systems offuzzy equations” Fuzzy Sets and Systems 12 (3) 301 - 302
Pedrycz W. 1993 ”s-t fuzzy relational equations” Fuzzy Sets and Systems 59 (2) 189 - 195
Perfilieva I. 2004 ”Fuzzy function as an approximate solutionto a system of fuzzy relation equations” Fuzzy Sets and Systems 147 (3) 363 - 383
Perfilieva I. , Noskova L. 2008 ”System of fuzzy relationequations with inf- composition: complete set of solutions” Fuzzy Sets and Systems 159 (17) 2256 - 2271
Bandler W. , Kohout L. J. 1980 ”Semantics of implicationoperators and fuzzy relational products” International Journal of Man-Machine Studies 12 (1) 89 - 116    DOI : 10.1016/S0020-7373(80)80055-1
Birkhoff G. 1967 Lattice Theory 3rd ed. American Mathematical Society Providence, RI
Dvurecenskij A. 2002 ”Pseudo MV-algebras are intervals inl-groups” Journal of the Australian Mathematical Society 72 (3) 427 - 445    DOI : 10.1017/S1446788700036806
Dvurecenskij A. 2001 ”On pseudo MV-algebras” Soft Computing 5 (5) 347 - 354    DOI : 10.1007/s005000100136
Galatos N. , Tsinakis C. 2005 ”Generalized MV-algebras” Journal of Algebra 283 (1) 254 - 291    DOI : 10.1016/j.jalgebra.2004.07.002
Georgescu G. , Leustean L. 2002 ”Some classes of pseudo-BL algebras” Journal of the Australian Mathematical Society 73 (1) 127 - 154    DOI : 10.1017/S144678870000851X
Georgescu G. , lorgulescu A. 2001 ”Pseudo MV-algebras” Multiple-Valued Logics 6 193 - 215
Georgescu G. , Popescu A. 2003 ”Non-commutativefuzzy Galois connections” Soft Computing 7 (7) 458 - 467
Georgescu G. , Popescu A. 2004 ”Non-commutative fuzzystructures and pairs of weak negations” Advances in Fuzzy Logic 143 (7) 129 - 155
Hoohle U. , Klement E. P. 1995 Non-Classical Logics andTheir Applications to Fuzzy Subsets: A Handbook of theMathematical Foundations of Fuzzy Set Theory Kluwer Academic Publishers Boston