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Special Types of Fuzzy Relations
Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 31 (2014) 552 – 557
2nd International Conference on Information Technology and Quantitative Management, ITQM 2014
Special Types of Fuzzy Relations S. N˘ad˘abana , I. Dzitaca,b,∗ a Aurel
Vlaicu University of Arad, Department of Mathematics and Computer Science, Elena Dr˘agoi 2, RO-310330 Arad, Romˆania. b Agora University of Oradea , Department of Social Sciences, Piat¸a Tineretului 8, RO-485526 Oradea, Romˆ ania.
Abstract The aim of this paper is to present, in an unitary way, some special types of fuzzy relations: affine fuzzy relations, linear fuzzy relations, convex fuzzy relations, M-convex fuzzy relations. All these fuzzy relations are characterized and we established the inclusions between these classes of fuzzy relations. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license c 2014 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection andpeer-review peer-reviewunder underresponsibility responsibilityofofthe theOrganizing Organizing Committee ITQM 2014. Selection and Committee of of ITQM 2014.
Keywords: fuzzy relations, fuzzy multifunctions, fuzzy multivalued mappings, convex fuzzy processes, affine fuzzy relations, linear fuzzy relations, convex fuzzy relations.
1. Introduction It is well known that fuzzy relations play an important role in fuzzy modeling and fuzzy control and they also have important applications in relational databases, approximate reasoning, preference modeling, medical diagnosis. The aim of this paper is to present, in an unitary way, some special types of fuzzy relations: affine fuzzy relations, linear fuzzy relations, convex fuzzy relations, M-convex fuzzy relations, in order to build a fertile ground for application, in further papers, of these fuzzy relations in decision making. All these fuzzy relations are characterized and we established the inclusions between these classes of fuzzy relations. In this paper X, Y will be vector spaces over K, where K is R or C. The concept of fuzzy relation was introduced by L.A. Zadeh in his classical paper 10 . According to L.A. Zadeh a fuzzy relation T between X and Y is a fuzzy set in X × Y, i.e. a mapping T : X × Y → [0, 1]. We denote by FR(X, Y) the family of all fuzzy relations between X and Y. For x ∈ X we denote by T x the fuzzy set in Y defined by T x (y) = T (x, y). Thus a fuzzy relation can be seen as a mapping X x → T x ∈ F (Y), where F (Y) represents the family of all fuzzy sets in Y. Such mappings were investigated by various mathematicians under different aspects. Thus N. Papageorgiou 6 called this mappings fuzzy multifunctions and studied the continuity of these mappings. E. Tsiporkova, B. De Baets, E. Kerre 9 called this maps fuzzy multivalued mappings and they defined lower and upper semi-continuous fuzzy ∗
Corresponding author. Tel.: +40-722-562053 ; Fax: +40-359-101032 . E-mail address: [email protected]
1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. doi:10.1016/j.procs.2014.05.301
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S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557
multivalued mapping and the relationships between these two types are studied completely. The continuity of fuzzy multifunctions was also studied by I. Beg 1 . Any application T : Rm → F (Rn ) will be called, by Y. Chalco-Cano, M.A. Rojas-Medar, R. Osuna-G´omez 2 a fuzzy process. A special attention was accorded to convex fuzzy processes. These was introduced by M. Matloka 5 in 2000. Another concept of convex fuzzy process was proposed by Y. Syau, C. Low and T. Wu 8 in 2002. A comparative study of these fuzzy convex processes was made in 2010 by D. Qiu, F. Yang, L. Shu 7 . To avoid any confusion D. Qiu, F. Yang and L. Shu called the former M-convex fuzzy process and the latter SLW-convex process. 2. Preliminary In this paper, the symbols ∨ and ∧ are used for the supremum and infimum of a family of fuzzy sets. We write μ1 ⊆ μ2 if μ1 (x) ≤ μ2 (x), (∀)x ∈ X. Definition 2.1. 3 Let μ1 , μ2 , · · · , μn be fuzzy sets in X. The sum of fuzzy sets μ1 , μ2 , · · · , μn is denoted by μ1 + μ2 + · · · + μn and it is defined by (μ1 + μ2 + · · · + μn )(x) = sup μ1 (x1 ) ∧ μ2 (x2 ) ∧ · · · ∧ μn (xn ) . x1 +x2 +···+xn =x
Let μ ∈ F (X) and λ ∈ K. The fuzzy set λμ is defined by ⎧ x ⎪ ⎪ if λ 0 ⎪ ⎪μ λ ⎨ (λμ)(x) = ⎪ 0 if λ = 0, x 0 . ⎪ ⎪ ⎪ ⎩ ∨{μ(y) : y ∈ X} if λ = 0, x = 0 Definition 2.2. 3 A fuzzy set μ ∈ F (X) is called linear fuzzy subspace of X if
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1. μ + μ ⊆ μ; 2. λμ ⊆ μ, (∀)λ ∈ K. We de note by LFS (X) the family of all linear fuzzy subspace of X. Proposition 2.3. 3 Let μ ∈ F (X). The following statements are equivalent: 1. μ ∈ LFS (X); 2. (∀)α, β ∈ K, we have αμ + βμ ⊆ μ; 3. (∀)x, y ∈ X, (∀)α, β ∈ K, we have μ(αx + βy) ≥ μ(x) ∧ μ(y). Proposition 2.4. 3 If μ, ρ ∈ LFS (X) and λ ∈ K, then: 1. μ + ρ ∈ LFS (X); 2. λμ ∈ LFS (X); 3. μ(x) ≤ μ(0), (∀)x ∈ X. Definition 2.5. 10 A fuzzy set μ ∈ F (X) is called convex fuzzy set if μ(λx1 + (1 − λ)x2 ) ≥ μ(x1 ) ∧ μ(x2 ), (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] . Definition 2.6. 4 A fuzzy set μ ∈ F (X) is called fuzzy cone if μ(αx) = μ(x), (∀)x ∈ X, (∀)α > 0 .
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A fuzzy set μ ∈ F (X) is called convex fuzzy cone if it is a fuzzy cone which is also a convex fuzzy set. 3. Affine Fuzzy Relations Definition 3.1. A fuzzy set μ ∈ F (X) is called affine fuzzy set if μ(λx1 + (1 − λ)x2 ) ≥ μ(x1 ) ∧ μ(x2 ), (∀)x1 , x2 ∈ X, (∀)λ ∈ K .
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Definition 3.2. A fuzzy relation T between X and Y is called affine fuzzy relation if it is an affine fuzzy set in X × Y. We denote by AFR(X, Y) the family of all affine fuzzy relations between X and Y. Remark 3.3. A fuzzy relation T between X and Y is an affine fuzzy relation if and only if T (λ(x1 , y1 ) + (1 − λ)(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ), (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y, (∀)λ ∈ K .
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Theorem 3.4. Let T ∈ FR(X, Y). Then T ∈ AFR(X, Y) if and only if λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ K .
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Proof. As
⎧ y ⎪ ⎪ if λ 0 ⎪ ⎪ Tx λ ⎨ (λT x )(y) = ⎪ 0 if λ = 0, y 0 , ⎪ ⎪ ⎪ ⎩ ∨{T x (z) : z ∈ Y} if λ = 0, y = 0
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we must consider the following cases: Case 1. λ 0, λ 1. Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
y y 1 2 = sup T x1 ∧ T x2 λ 1 − λ y1 +y2 =y y1 y2 = sup T x1 , ∧ T x2 , 1 − λ y1 +y2 =y λ y1 y2 ≤ sup T λ x1 , + (1 − λ) x2 , λ 1−λ y1 +y2 =y = sup T ((λx1 , y1 ) + ((1 − λ)x2 , y2 ))
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y1 +y2 =y
= sup T (λx1 + (1 − λ)x2 , y1 + y2 ) y1 +y2 =y
= T (λx1 + (1 − λ)x2 , y) = T λx1 +(1−λ)x2 (y) Thus λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 . Case 2. λ = 0. Let y ∈ Y. (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
= (λT x1 )(0) ∧ ((1 − λ)T x2 )(y) = sup T x1 (z) ∧ T x2 (y)
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z∈Y
≤ T x2 (y) = T λx1 +(1−λ)x2 (y) Case 3. λ = 1. The proof is similar to the one of previous case. 4. Linear Fuzzy Relations Definition 4.1. A fuzzy relation T ∈ FR(X, Y) is called linear fuzzy relation if it is a linear fuzzy subspace in X × Y. We denote by LFR(X, Y) the family of all linear fuzzy relations between X and Y. Remark 4.2. The linearity of T ∈ LFR(X, Y) can be written T (α(x1 , y1 ) + β(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ) , (∀)α, β ∈ K, (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y .
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S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557
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We also note that, for β = 0, we obtain T (α(x1 , y1 )) ≥ T (x1 , y1 ) , (∀)α ∈ K, (∀)(x1 , y1 ) ∈ X × Y . If α 0, we have T (x1 , y1 ) = T
1 (αx1 , αy1 ) ≥ T (αx1 , αy1 ) = T (α(x1 , y1 )) . α
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Thus T (α(x1 , y1 )) = T (x1 , y1 ) , (∀)α ∈ K∗ , (∀)(x1 , y1 ) ∈ X × Y .
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Remark 4.3. LFR(X, Y) ⊂ AFR(X, Y). Theorem 4.4. Let T ∈ FR(X, Y). The following statements are equivalent: 1. T ∈ LFR(X, Y); 2. (a) T x1 + T x2 ⊆ T x1 +x2 , (∀)x1 , x2 ∈ X; (b) αT x ⊆ T αx , (∀)x ∈ X, (∀)α ∈ K; 3. αT x1 + βT x2 ⊆ T αx1 +βx2 , (∀)x1 , x2 ∈ X, (∀)α, β ∈ K. Proof. It is similar to the proof of Theorem 3.4. 5. Convex Fuzzy Relations Definition 5.1. A fuzzy relation T ∈ FR(X, Y) is called convex fuzzy relation if it is a convex fuzzy set in X × Y. We denote by CFR(X, Y) the family of all convex fuzzy relations between X and Y. Remark 5.2. Let T ∈ FR(X, Y). Then T is a convex fuzzy relation if and only if T (λ(x1 , y1 ) + (1 − λ)(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ) , (∀)λ ∈ [0, 1], (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y .
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Remark 5.3. AFR(X, Y) ⊂ CFR(X, Y). Theorem 5.4. Let T ∈ FR(X, Y). Then T ∈ CFR(X, Y) if and only if λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] .
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Proof. ” ⇒ ” See 7 . ” ⇐ ” See 8 . Remark 5.5. The definition of convex fuzzy relation it’s not new. This maps were called by Y. Syau, C. Low and T. Wu 8 convex fuzzy processes. 6. M-Convex Fuzzy Relations Definition 6.1. A fuzzy relation T ∈ FR(X, Y) is called M-convex fuzzy relation if it is a convex fuzzy cone in X × Y. We denote by MCFR(X, Y) the family of all M-convex fuzzy relations between X and Y. Theorem 6.2. Let T ∈ FR(X, Y). Then T ∈ MCFR(X, Y) if and only if 1. T x1 + T x2 ⊆ T x1 +x2 , (∀)x1 , x2 ∈ X; 2. αT x = T αx , (∀)x ∈ X, (∀)α > 0;
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S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557
Proof. ” ⇒ ” See 7 . ” ⇐ ” See 5 . Remark 6.3. LFR(X, Y) ⊂ MCFR(X, Y). Indeed, we must show that αT x = T αx , (∀)x ∈ X, (∀)α > 0. We have y y 1 αT x (y) = T x = T x, = T (αx, y) = T (αx, y) = T αx (y) . α α α
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Remark 6.4. MCFR(X, Y) ⊂ CFR(X, Y). Indeed, we must show that. λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] .
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Case 1. λ = 0. Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = (0T x1 + T x2 )(y) = sup [(0T x1 )(y1 ) ∧ T x2 (y2 )] y1 +y2 =y
= (0T x1 )(0) ∧ T x2 (y) ≤ T x2 (y) = T 0x1 +1x2 (y)
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Case 2. λ = 1. The proof is similar to the one of previous case. Case 3. λ ∈ (0, 1). Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
= sup [T λx1 (y1 ) ∧ T (1−λ)x2 (y2 )] y1 +y2 =y
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= (T λx1 + T (1−λ)x2 )(y) ≤ T λx1 +(1−λ)x2 (y) Remark 6.5. The definition of M-convex fuzzy relation it’s not new. M. Matloka 5 introduced the concept of convex fuzzy process. Previous theorem shows that M-convex fuzzy relations are in fact convex fuzzy processes. Thus, for these types of fuzzy relations, we use the name M-convex fuzzy relations. 7. Conclusions and Future Works In this paper we have build a fertile ground to study, in further papers, special types of closed fuzzy relations between topological vector spaces. The results obtained in this paper leave to foresee that there are solutions to the problem afore mentioned. These fuzzy relations can be proven to be a powerful tool for decision making. Acknowledgments This work was supported in part by research centers: 1) SC Cercetare Dezvoltare Agora (R & D Agora) of Agora University of Oradea and 2) Mathematical Models and Information Systems, Faculty of Exact Sciences of Aurel Vlaicu University of Arad. References 1. Beg I. Continuity of fuzzy multifunctions. Journal of Applied Mathematics and Stochastic Analysis 1999;12:17-22. 2. Chalco-Cano Y, Rojas-Medar MA, Osuna-G´omez R. s-Convex fuzzy processes. Computers and Mathematics with Applications 2004;47:14111418.
S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557 3. Katsaras AK, Liu DB. Fuzzy vector spaces and fuzzy topological vector spaces. Journal of Mathematical Analysis and Applications 1977;58:135-146. 4. Liu YM. Some properties of convex fuzzy sets. Journal of Mathematical Analysis and Applications 1985;111:119-129. 5. Matloka M. Convex fuzzy processes. Fuzzy Sets and Systems 2000;110:109-114. 6. Papageorgiou NS. Fuzzy topology and fuzzy multifunctions. Journal of Mathematical Analysis and Applications 1985;109:397-425. 7. Qiu D, Yang F, Shu L. On convex fuzzy processes and their generalizations. International Journal of Fuzzy Systems 2010;12:268-273. 8. Syau YR, Low CY, Wu TH. A note on convex fuzzy processes. Applied Mathematics Letters 2002;15:193-196. 9. Tsiporkova E, De Baets B, Kerre E. Continuity of fuzzy multivalued mapping. Fuzzy Sets and Systems 1998;94:335-348. 10. Zadeh LA. Fuzzy Sets. Informations and Control 1965;8:338-353.
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ScienceDirect Procedia Computer Science 31 (2014) 552 – 557
2nd International Conference on Information Technology and Quantitative Management, ITQM 2014
Special Types of Fuzzy Relations S. N˘ad˘abana , I. Dzitaca,b,∗ a Aurel
Vlaicu University of Arad, Department of Mathematics and Computer Science, Elena Dr˘agoi 2, RO-310330 Arad, Romˆania. b Agora University of Oradea , Department of Social Sciences, Piat¸a Tineretului 8, RO-485526 Oradea, Romˆ ania.
Abstract The aim of this paper is to present, in an unitary way, some special types of fuzzy relations: affine fuzzy relations, linear fuzzy relations, convex fuzzy relations, M-convex fuzzy relations. All these fuzzy relations are characterized and we established the inclusions between these classes of fuzzy relations. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license c 2014 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection andpeer-review peer-reviewunder underresponsibility responsibilityofofthe theOrganizing Organizing Committee ITQM 2014. Selection and Committee of of ITQM 2014.
Keywords: fuzzy relations, fuzzy multifunctions, fuzzy multivalued mappings, convex fuzzy processes, affine fuzzy relations, linear fuzzy relations, convex fuzzy relations.
1. Introduction It is well known that fuzzy relations play an important role in fuzzy modeling and fuzzy control and they also have important applications in relational databases, approximate reasoning, preference modeling, medical diagnosis. The aim of this paper is to present, in an unitary way, some special types of fuzzy relations: affine fuzzy relations, linear fuzzy relations, convex fuzzy relations, M-convex fuzzy relations, in order to build a fertile ground for application, in further papers, of these fuzzy relations in decision making. All these fuzzy relations are characterized and we established the inclusions between these classes of fuzzy relations. In this paper X, Y will be vector spaces over K, where K is R or C. The concept of fuzzy relation was introduced by L.A. Zadeh in his classical paper 10 . According to L.A. Zadeh a fuzzy relation T between X and Y is a fuzzy set in X × Y, i.e. a mapping T : X × Y → [0, 1]. We denote by FR(X, Y) the family of all fuzzy relations between X and Y. For x ∈ X we denote by T x the fuzzy set in Y defined by T x (y) = T (x, y). Thus a fuzzy relation can be seen as a mapping X x → T x ∈ F (Y), where F (Y) represents the family of all fuzzy sets in Y. Such mappings were investigated by various mathematicians under different aspects. Thus N. Papageorgiou 6 called this mappings fuzzy multifunctions and studied the continuity of these mappings. E. Tsiporkova, B. De Baets, E. Kerre 9 called this maps fuzzy multivalued mappings and they defined lower and upper semi-continuous fuzzy ∗
Corresponding author. Tel.: +40-722-562053 ; Fax: +40-359-101032 . E-mail address: [email protected]
1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. doi:10.1016/j.procs.2014.05.301
553
S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557
multivalued mapping and the relationships between these two types are studied completely. The continuity of fuzzy multifunctions was also studied by I. Beg 1 . Any application T : Rm → F (Rn ) will be called, by Y. Chalco-Cano, M.A. Rojas-Medar, R. Osuna-G´omez 2 a fuzzy process. A special attention was accorded to convex fuzzy processes. These was introduced by M. Matloka 5 in 2000. Another concept of convex fuzzy process was proposed by Y. Syau, C. Low and T. Wu 8 in 2002. A comparative study of these fuzzy convex processes was made in 2010 by D. Qiu, F. Yang, L. Shu 7 . To avoid any confusion D. Qiu, F. Yang and L. Shu called the former M-convex fuzzy process and the latter SLW-convex process. 2. Preliminary In this paper, the symbols ∨ and ∧ are used for the supremum and infimum of a family of fuzzy sets. We write μ1 ⊆ μ2 if μ1 (x) ≤ μ2 (x), (∀)x ∈ X. Definition 2.1. 3 Let μ1 , μ2 , · · · , μn be fuzzy sets in X. The sum of fuzzy sets μ1 , μ2 , · · · , μn is denoted by μ1 + μ2 + · · · + μn and it is defined by (μ1 + μ2 + · · · + μn )(x) = sup μ1 (x1 ) ∧ μ2 (x2 ) ∧ · · · ∧ μn (xn ) . x1 +x2 +···+xn =x
Let μ ∈ F (X) and λ ∈ K. The fuzzy set λμ is defined by ⎧ x ⎪ ⎪ if λ 0 ⎪ ⎪μ λ ⎨ (λμ)(x) = ⎪ 0 if λ = 0, x 0 . ⎪ ⎪ ⎪ ⎩ ∨{μ(y) : y ∈ X} if λ = 0, x = 0 Definition 2.2. 3 A fuzzy set μ ∈ F (X) is called linear fuzzy subspace of X if
(1)
(2)
1. μ + μ ⊆ μ; 2. λμ ⊆ μ, (∀)λ ∈ K. We de note by LFS (X) the family of all linear fuzzy subspace of X. Proposition 2.3. 3 Let μ ∈ F (X). The following statements are equivalent: 1. μ ∈ LFS (X); 2. (∀)α, β ∈ K, we have αμ + βμ ⊆ μ; 3. (∀)x, y ∈ X, (∀)α, β ∈ K, we have μ(αx + βy) ≥ μ(x) ∧ μ(y). Proposition 2.4. 3 If μ, ρ ∈ LFS (X) and λ ∈ K, then: 1. μ + ρ ∈ LFS (X); 2. λμ ∈ LFS (X); 3. μ(x) ≤ μ(0), (∀)x ∈ X. Definition 2.5. 10 A fuzzy set μ ∈ F (X) is called convex fuzzy set if μ(λx1 + (1 − λ)x2 ) ≥ μ(x1 ) ∧ μ(x2 ), (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] . Definition 2.6. 4 A fuzzy set μ ∈ F (X) is called fuzzy cone if μ(αx) = μ(x), (∀)x ∈ X, (∀)α > 0 .
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A fuzzy set μ ∈ F (X) is called convex fuzzy cone if it is a fuzzy cone which is also a convex fuzzy set. 3. Affine Fuzzy Relations Definition 3.1. A fuzzy set μ ∈ F (X) is called affine fuzzy set if μ(λx1 + (1 − λ)x2 ) ≥ μ(x1 ) ∧ μ(x2 ), (∀)x1 , x2 ∈ X, (∀)λ ∈ K .
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Definition 3.2. A fuzzy relation T between X and Y is called affine fuzzy relation if it is an affine fuzzy set in X × Y. We denote by AFR(X, Y) the family of all affine fuzzy relations between X and Y. Remark 3.3. A fuzzy relation T between X and Y is an affine fuzzy relation if and only if T (λ(x1 , y1 ) + (1 − λ)(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ), (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y, (∀)λ ∈ K .
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Theorem 3.4. Let T ∈ FR(X, Y). Then T ∈ AFR(X, Y) if and only if λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ K .
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Proof. As
⎧ y ⎪ ⎪ if λ 0 ⎪ ⎪ Tx λ ⎨ (λT x )(y) = ⎪ 0 if λ = 0, y 0 , ⎪ ⎪ ⎪ ⎩ ∨{T x (z) : z ∈ Y} if λ = 0, y = 0
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we must consider the following cases: Case 1. λ 0, λ 1. Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
y y 1 2 = sup T x1 ∧ T x2 λ 1 − λ y1 +y2 =y y1 y2 = sup T x1 , ∧ T x2 , 1 − λ y1 +y2 =y λ y1 y2 ≤ sup T λ x1 , + (1 − λ) x2 , λ 1−λ y1 +y2 =y = sup T ((λx1 , y1 ) + ((1 − λ)x2 , y2 ))
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y1 +y2 =y
= sup T (λx1 + (1 − λ)x2 , y1 + y2 ) y1 +y2 =y
= T (λx1 + (1 − λ)x2 , y) = T λx1 +(1−λ)x2 (y) Thus λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 . Case 2. λ = 0. Let y ∈ Y. (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
= (λT x1 )(0) ∧ ((1 − λ)T x2 )(y) = sup T x1 (z) ∧ T x2 (y)
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z∈Y
≤ T x2 (y) = T λx1 +(1−λ)x2 (y) Case 3. λ = 1. The proof is similar to the one of previous case. 4. Linear Fuzzy Relations Definition 4.1. A fuzzy relation T ∈ FR(X, Y) is called linear fuzzy relation if it is a linear fuzzy subspace in X × Y. We denote by LFR(X, Y) the family of all linear fuzzy relations between X and Y. Remark 4.2. The linearity of T ∈ LFR(X, Y) can be written T (α(x1 , y1 ) + β(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ) , (∀)α, β ∈ K, (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y .
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S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557
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We also note that, for β = 0, we obtain T (α(x1 , y1 )) ≥ T (x1 , y1 ) , (∀)α ∈ K, (∀)(x1 , y1 ) ∈ X × Y . If α 0, we have T (x1 , y1 ) = T
1 (αx1 , αy1 ) ≥ T (αx1 , αy1 ) = T (α(x1 , y1 )) . α
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Thus T (α(x1 , y1 )) = T (x1 , y1 ) , (∀)α ∈ K∗ , (∀)(x1 , y1 ) ∈ X × Y .
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Remark 4.3. LFR(X, Y) ⊂ AFR(X, Y). Theorem 4.4. Let T ∈ FR(X, Y). The following statements are equivalent: 1. T ∈ LFR(X, Y); 2. (a) T x1 + T x2 ⊆ T x1 +x2 , (∀)x1 , x2 ∈ X; (b) αT x ⊆ T αx , (∀)x ∈ X, (∀)α ∈ K; 3. αT x1 + βT x2 ⊆ T αx1 +βx2 , (∀)x1 , x2 ∈ X, (∀)α, β ∈ K. Proof. It is similar to the proof of Theorem 3.4. 5. Convex Fuzzy Relations Definition 5.1. A fuzzy relation T ∈ FR(X, Y) is called convex fuzzy relation if it is a convex fuzzy set in X × Y. We denote by CFR(X, Y) the family of all convex fuzzy relations between X and Y. Remark 5.2. Let T ∈ FR(X, Y). Then T is a convex fuzzy relation if and only if T (λ(x1 , y1 ) + (1 − λ)(x2 , y2 )) ≥ T (x1 , y1 ) ∧ T (x2 , y2 ) , (∀)λ ∈ [0, 1], (∀)(x1 , y1 ), (x2 , y2 ) ∈ X × Y .
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Remark 5.3. AFR(X, Y) ⊂ CFR(X, Y). Theorem 5.4. Let T ∈ FR(X, Y). Then T ∈ CFR(X, Y) if and only if λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] .
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Proof. ” ⇒ ” See 7 . ” ⇐ ” See 8 . Remark 5.5. The definition of convex fuzzy relation it’s not new. This maps were called by Y. Syau, C. Low and T. Wu 8 convex fuzzy processes. 6. M-Convex Fuzzy Relations Definition 6.1. A fuzzy relation T ∈ FR(X, Y) is called M-convex fuzzy relation if it is a convex fuzzy cone in X × Y. We denote by MCFR(X, Y) the family of all M-convex fuzzy relations between X and Y. Theorem 6.2. Let T ∈ FR(X, Y). Then T ∈ MCFR(X, Y) if and only if 1. T x1 + T x2 ⊆ T x1 +x2 , (∀)x1 , x2 ∈ X; 2. αT x = T αx , (∀)x ∈ X, (∀)α > 0;
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Proof. ” ⇒ ” See 7 . ” ⇐ ” See 5 . Remark 6.3. LFR(X, Y) ⊂ MCFR(X, Y). Indeed, we must show that αT x = T αx , (∀)x ∈ X, (∀)α > 0. We have y y 1 αT x (y) = T x = T x, = T (αx, y) = T (αx, y) = T αx (y) . α α α
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Remark 6.4. MCFR(X, Y) ⊂ CFR(X, Y). Indeed, we must show that. λT x1 + (1 − λ)T x2 ⊆ T λx1 +(1−λ)x2 , (∀)x1 , x2 ∈ X, (∀)λ ∈ [0, 1] .
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Case 1. λ = 0. Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = (0T x1 + T x2 )(y) = sup [(0T x1 )(y1 ) ∧ T x2 (y2 )] y1 +y2 =y
= (0T x1 )(0) ∧ T x2 (y) ≤ T x2 (y) = T 0x1 +1x2 (y)
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Case 2. λ = 1. The proof is similar to the one of previous case. Case 3. λ ∈ (0, 1). Let y ∈ Y. Then (λT x1 + (1 − λ)T x2 )(y) = sup [(λT x1 )(y1 ) ∧ ((1 − λ)T x2 )(y2 )] y1 +y2 =y
= sup [T λx1 (y1 ) ∧ T (1−λ)x2 (y2 )] y1 +y2 =y
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= (T λx1 + T (1−λ)x2 )(y) ≤ T λx1 +(1−λ)x2 (y) Remark 6.5. The definition of M-convex fuzzy relation it’s not new. M. Matloka 5 introduced the concept of convex fuzzy process. Previous theorem shows that M-convex fuzzy relations are in fact convex fuzzy processes. Thus, for these types of fuzzy relations, we use the name M-convex fuzzy relations. 7. Conclusions and Future Works In this paper we have build a fertile ground to study, in further papers, special types of closed fuzzy relations between topological vector spaces. The results obtained in this paper leave to foresee that there are solutions to the problem afore mentioned. These fuzzy relations can be proven to be a powerful tool for decision making. Acknowledgments This work was supported in part by research centers: 1) SC Cercetare Dezvoltare Agora (R & D Agora) of Agora University of Oradea and 2) Mathematical Models and Information Systems, Faculty of Exact Sciences of Aurel Vlaicu University of Arad. References 1. Beg I. Continuity of fuzzy multifunctions. Journal of Applied Mathematics and Stochastic Analysis 1999;12:17-22. 2. Chalco-Cano Y, Rojas-Medar MA, Osuna-G´omez R. s-Convex fuzzy processes. Computers and Mathematics with Applications 2004;47:14111418.
S. Nădăban and I. Dzitac / Procedia Computer Science 31 (2014) 552 – 557 3. Katsaras AK, Liu DB. Fuzzy vector spaces and fuzzy topological vector spaces. Journal of Mathematical Analysis and Applications 1977;58:135-146. 4. Liu YM. Some properties of convex fuzzy sets. Journal of Mathematical Analysis and Applications 1985;111:119-129. 5. Matloka M. Convex fuzzy processes. Fuzzy Sets and Systems 2000;110:109-114. 6. Papageorgiou NS. Fuzzy topology and fuzzy multifunctions. Journal of Mathematical Analysis and Applications 1985;109:397-425. 7. Qiu D, Yang F, Shu L. On convex fuzzy processes and their generalizations. International Journal of Fuzzy Systems 2010;12:268-273. 8. Syau YR, Low CY, Wu TH. A note on convex fuzzy processes. Applied Mathematics Letters 2002;15:193-196. 9. Tsiporkova E, De Baets B, Kerre E. Continuity of fuzzy multivalued mapping. Fuzzy Sets and Systems 1998;94:335-348. 10. Zadeh LA. Fuzzy Sets. Informations and Control 1965;8:338-353.
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