International Mathematical Forum, 5, 2010, no. 8, 373-381 The Properties of Fuzzy Relations Yong Chan Kim Department of Mathematics, Gangneung-Wonju National University Gangneung, Gangwondo 210-702, Korea yck@gwnu.ac.kr Young Sun Kim Department of Applied Mathematics, Pai Chai University Dae Jeon, 302-735, Korea yskim@pcu.ac.kr Abstract We investigate the relations between fuzzy relations and maps on fuzzy sets. Moreover, we give their examples. Mathematics Subject Classification: 03E72, 03G10, 06A15, 06F07 Keywords: Complete residuated lattice, Fuzzy relations, -equivalence relation 1 Introduction Jacas and Recasens [7] introduced the notion of fuzzy T-equivalence relation on t-norm. Recently, Höhle [5,6] developed the algebraic structures and many valued topologies in a sense of quantales and cqm-lattices. Bělohlávek [1-3] investigate the properties of fuzzy relations and similarities on a residual lattice which supports part of foundation of theoretic computer science. In this paper, we investigate the relationships between fuzzy relations L ) and maps on fuzzy sets L ) L ) on a complete residuated lattice. Moreover, we give their examples. 2 Preliminaries Definition 2.1 [1-3,9] A triple L,, ) is called a complete residuated lattice iff it satisfies the following conditions:
374 Yong Chan Kim and Young Sun Kim L1) L =L,, 1, 0) is a complete lattice where 1 is the universal upper bound and 0 denotes the universal lower bound; L2) L,, 1) is a commutative monoid; L3) is distributive over arbitrary joins, i.e. a i ) b = i b). i Γ i Γa Define an operation as a b = {c L a c b}, for each a, b L. Example 2.2 [1-3,9] 1) Each frame L,, ) is a complete residuated lattice. 2) The unit interval with a left-continuous t-norm t, [0, 1],,t), is a complete residuated lattice. 3) Define a binary operation on [0, 1] by x y = max{0,x+ y 1}. Then [0, 1],, ) is a complete residuated lattice. Let L,, ) be a complete residuated lattice. A order reversing map : L L defined by a = a 0 is called a strong negation if a = a for each a L. Lemma 2.3 [1-3,9] Let L,, ) be a complete residuated lattice with a strong negation. For each x, y, z, x i,y i L, we have the following properties. 1) If y z, x y) x z), x y x z and z x y x. 2) x y x y. 3) x i Γ y i )= i Γx y i ). 4) i Γ x i ) y = i Γx i y). 5) x i Γ y i ) i Γx y i ). 6) i Γ x i ) y i Γx i y). 7) x y) z = x y z) =y x z). 8) x x y) y and x x y) y. 9) y x x y) and x x y z) y z. 10) x y) y z) x z. 11) x y =1iffx y. Definition 2.4 [1-3,7,9] Let be a set. A function R : L is called: R1) reflexive if Rx, x) = 1 for all x, R2) symmetric if Rx, y) =Ry, x), for all x, y, R3) transitive if Rx, y) Ry, z) Rx, z), for all x, y, z. If R satisfies R1) and R2), R is an -quasi-equivalence relation. If an -quasi-equivalence relation satisfies R2), R is an an -equivalence relation.
The properties of fuzzy relations 375 3 The Properties of Fuzzy Relations In this section, we investigate the relationships between L and L ) L on a complete residuated lattice L,,, ) with a strong negation. Theorem 3.1 We define a mapping Φ:L L ) L as follows: ΦR)λ)y) = λx) Rx, y). x Then we have the following properties: 1) Φ has a right adjoint mapping Λ:L ) L L as follows: Λφ )x, y) = λ L λx) φ λ)y) ). 2) ΦΛφ )) φ and Λ Φ=1 L Y. 3) Let RL )={R L Rx, x) =1, x } and E, ) = {φ L ) L λ φ λ), λ L }. Then Φ:RL ) E, ) and Λ:E, ) RL ) are well defined. 4) Let T L )={R L y Rx, y) Ry, z) Rx, z), x, z } and T, ) ={φ L ) L φ φ φ }. Then Φ:T L ) T, ) and Λ:T, ) T L ) are well defined. Proof. 1) Φ is join preserving because Φ i R i )λ)y) = x λx) i R i x, y) = i x λx) R i x, y)) = i ΦR i )λ)y). Hence Φ has a right adjoint mapping Λ as follows: Λφ )x, y) = {Rx, y) ΦR)λ)y) φ λ)y)} = {Rx, y) λx) Rx, y) φ λ)y)} = {Rx, y) Rx, y) λ L λx) φ λ)y) ) } = λ L λx) φ λ)y) ). 2) ΦΛφ ))λ)y) = x λx) Λφ )x, y) ) = x λx) ρ L ρx) φ ρ)y) )) x λx) λx) φ λ)y) )) φ λ)y).
376 Yong Chan Kim and Young Sun Kim ΛΦR))x, y) = λ Lλx) ΦR)λ)y)) = λ L λx) z λz) Rz, y)) ) ) λ L λx) λx) Rx, y)) Rx, y). Hence Λ Φ 1 L Y. Moreover, Λ Φ 1 L Y ΛΦR))x, y) 3) It easily proved. 4) from: = λ L λx) z λz) Rz, y)) ) 1 x x) z 1 x z) Rz, y)) ) =1 1 Rx, y)) = Rx, y). ΦR)ΦR))λ)y) = z ΦR)λ)z) Rz, y) = z x λx) Rx, z) ) Rz, y) = x λx) z Rx, z) Rz, y)) x λx) Rx, y)) = Φλ)y). Λφ )x, y) Λφ )y, z) = λ L λx) φ λ)y) ) ρ L ρy) φ ρ)z) ) λ L λx) φ λ)y) ) φ λ)y) φ φ λ))z) ) λ L λx) φ φ λ))z) ) λ L λx) φ λ)z) ) =Λφ )x, z). Theorem 3.2 We define a mapping Ψ:L L ) L ΨR)λ)y) = Rx, y) λx)). x as follows: Then we have the following properties: 1) Ψ has a right adjoint mapping Γ:L ) L L as follows: Γφ )x, y) = λ L 2) Ψ Γ 1 L ) L and Γ Ψ=1 L. φ λ)y) λx) ).
The properties of fuzzy relations 377 3) Let RL )={R L Rx, x) =1, x } and F, ) = {φ L ) L φ λ) λ, λ L }. Then Ψ:RL ) F, ) and Γ:F, ) RL ) are well defined. 4) Let T L )={R L y Rx, y) Ry, z) Rx, z), x, z } and I, ) ={φ L ) L φ φ φ }. Then Ψ:T L ) I, ) and Γ:I, ) T L ) are well defined. Proof. 1) Since Ψ i Γ R i )λ)y) = i Γ ΨR i )λ)y) = op i Γ ΨR i )λ)y), Ψ has a right adjoint mapping Γ with op = as follows: Γφ )x, y) = {Rx, y) ΨR)λ)y) op φ λ)y)} = {Rx, y) ΦR)λ)y) φ λ)y)} = {Rx, y) φ λ)y) Rx, y) λx)} = {Rx, y) Rx, y) λ L φ λ)y) λx)} = λ L φ λ)y) λx). 2) We have Ψ Γ 1 L ) L from ΨΓφ ))λ)y) = x Γφ )x, y) λx)) = x ρ L φ ρ)y) ρx)) λx) ) x φ λ)y) λx)) λx) ) φ λ)y). by Lemma 2.38)) We have Γ Φ 1 L Y from ΓΨR))x, y) = ) λ L ΨR)λ)y) λx) = λ L z Rz, y) λz)) λx) ) ) λ L Rx, y) λx)) λx) Rx, y). Put λ =1 x. Then ΓΨR))x, y) = λ L z Rz, y) λz)) λx) ) 3) It easily proved. 4) ΦΦR))λ)y) Rx, y) 1 x x)) 1 x x)) =Rx, y) 0) 0=Rx, y). = z Rz, y) ΦR)λ)z)) = z Rz, y) x Rx, z) λx)) = z x Rz, y) Rx, z) λx)) = x z Rz, y) Rx, z)) λx)) by Lemma 2.37)) = x z Rz, y) Rx, z)) λx)) x Rx, y) λx)) =ΦR)λ)y).
378 Yong Chan Kim and Young Sun Kim Γφ )x, y) Γφ )y, z) = λ L φ λ)y) λx) ) ρ L φ ρ)z) ρy) ) λ L φ λ)y) λx) ) φ φ λ))z) φ λ)y) ) λ L φ φ λ))z) λx) ) by Lemma 2.312)) λ L φ λ)z) λx) ) =Γφ )x, z). Theorem 3.3 We define a mapping Ω:L L ) L as follows: ΩR)λ)y) = λx) Rx, y)). x Then we have the following properties: 1) Ω has a left adjoint mapping Υ:L ) L L as follows: Υω )x, y) = λx) ω λ)y) ). λ L 2) Ω Υ 1 L ) L and Υ Ω=1 L. 3) Let RL )={R L Rx, x) =1} and G, ) ={φ L ) L λ x L λ xx) φ λ x )x) =1, x }. Then Ω:RL ) G, ) and Υ:G, ) RL ) are well defined. Proof. 1) Since Ω i Γ R i )λ)y) = i Γ ΩR i )λ)y), Ω has a left adjoint mapping Υ as follows: 2) Υω )x, y) = {Rx, y) ΩR)λ)y) ω λ)y), λ L } = {Rx, y) ω λ)y) λx) Rx, y), λ L } = {Rx, y) λx) ω λ)y) Rx, y), λ L } = λ L λx) ω λ)y). ΩΥω ))λ)y) = x λx) Υω )x, y)) = x λx) λ L λx) ω λ)y) ) x λx) λx) ω λ)y) ) = ω λ)y). Hence Ω Υ 1 L ) L. ΥΩR))x, y) = λ L λx) ΩR)λ)y) ) = λ L λx) z λz) Rz, y)) ) λ L λx) λx) Rx, y)) ) Rx, y). by Lemma 2.38))
The properties of fuzzy relations 379 Put λ =1 x. ΥΩR))x, y) = λ L λx) z λz) Rz, y)) ) λ L 1x x) z 1 x z) Rz, y)) ) = Rx, y). Hence Υ Ω=1 L. 3) Since 1 x x) ΩR)1 x )x) = 1 for R RL ) and x, ΩR) G, ). For ω G, ), Υω )x, x) = λ x L λ xx) ω λ x )x)) = 1. Hence Υω ) RL ). Example 3.4 Let [0, 1], ) be a complete residuated lattice defined as x y =x+y 1) 0. We obtain x y =1 x+y)1. Let = {x 1,x 2,x 3 } be a set and ρx 1 )=0.6,ρx 2 )=0.8,ρx 3 )=0.5. 1) We define φ ρ : L L as follows: φ ρ λ) = 1 if λ = 1, ρ if 1 λ ρ, 0 otherwise. Since φ ρ λ) λ and φ ρ φ ρ λ)) = φ ρ λ) for all λ L, by Theorem 3.23), φ ρ F, ),I, ). Since Γφ ρ )x, y) = λ L φ ρ λ)y) λx) ),we obtain a reflexive and transitive matrix Γφ ρ ) as follows: In general, Ψ Γ 1 L Since ΨΓφ ρ )) Λφ ρ )x, y) = Υφ ρ λx 1 ) λx 2 ) λx 3 ) )x, y) = λ L Λφ ρ )= Γφ ρ )= λ L because = 1 0.8 1 0.9 0.7 1 λx 1 ) λx 2 ) 0.1+λx 3 )) 0.2+λx 1 )) λx 2 ) 0.3+λx 3 )) λx 1 ) λx 2 ) λx 3 ) λx) φ ρ λ)y) ) =1 x x) φ ρ 1 x )y) =0, λx) φ ρ λ)y) ) = 1x) φ ρ Υφ ρ )= 1)y) =1,
380 Yong Chan Kim and Young Sun Kim 2) We define ψ ρ : L L as follows: ψ ρ λ) = 0 if λ = 0, ρ if 0 λ ρ, 1 otherwise. Since ψρ λ) λ and ψ ρ ψ ρ λ)) = ψ ρ λ) for all λ L, by Theorem 3.13,4), φ ρ E, ),T, ). Since Λψρ )x, y) = λ L λx) ψρ λ)y)), we obtain a reflexive and transitive matrix Λψρ ) as follows: In general, Φ Λ 1 L Since ΦΛψ ρ )) Γψ ρ )x, y) = λx 1 ) λx 2 ) λx 3 ) λ L Λψ ρ )= because = 1 1 0.9 0.8 1 0.7 λx 1 ) λx 2 ) 0.2) λx 3 ) λx 1 ) λx 2 ) λx 3 ) λx 1 ) 0.1) λx 2 ) 0.3) λx 3 ) ψ ρ λ)y) λx) ) = ψ ρ 1 y )y) 1 y x) =0, x y, Γψρ )x, x) =ψ ρ 1 y)x) 1 y x) =0, x y, Υψρ )x, y) = λx) ψ ρ λ)y) ) = 1x) ψρ 1)y) =1, Γψ ρ )= λ L Υψ ρ )= 3) We define ω : L L as follows: ω λ) = { ρi if λ = ρ i, 0 otherwise. where ρ 1 x 1 )=1,ρ 1 x 2 )=0.8,ρ 1 x 3 )=0.7, ρ 2 x 1 )=0.6,ρ 2 x 2 )=1,ρx 3 )= 0.9 and ρ 3 x 1 )=1,ρ 3 x 2 )=0.8,ρx 3 ) = 1. We obtain a reflexive and transitive matrix Υω )= 1 0.8 1 0.8 1 0.9 1 0.9 1
The properties of fuzzy relations 381 Moreover, ΩΥω )) Λω )= λx 1 ) λx 2 ) λx 3 ) = Γω )= 2 λx 1 )) 1.8 λx 2 )) 2 λx 3 )) 1 1.8 λx 1 )) 2 λx 2 )) 1.9 λx 3 )) 1 2 λx 1 )) 1.9 λx 2 )) 2 λx 3 )) 1 References [1] R. Bělohlávek, Similarity relations in concept lattices, J. Logic and Computation, 106) 2000), 823-845. [2] R. Bělohlávek, Fuzzy equational logic, Arch. Math. Log., 41 2002), 83-90. [3] R. Bělohlávek, Similarity relations and BK-relational products, Information Sciences, 126 2000), 287-295. [4] P. Hájek, Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht 1998). [5] U. Höhle, Many valued topology and its applications, Kluwer Academic Publisher, Boston, 2001). [6] U. Höhle, E. P. Klement, Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publisher, Boston, 1995. [7] J. Jacas, J. Recasens, Fuzzy T-transitive relations: eigenvectors and generators, Fuzzy Sets and Systems, 72 1995), 147-154. [8] S. E. Rodabaugh, E. P. Klement, Toplogical And Algebraic Structures In Fuzzy Sets, The Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic 20, Kluwer Academic Publishers, Boston/Dordrecht/London) 2003). [9] E. Turunen, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., 1999. Received: September, 2009