1. Special Types of Fuzzy Relations S. Nadaban*, I. Dzițac*,** *Aurel Vlaicu University of Arad, Department of Mathematics and Computer Science **Agora University of Oradea, Department of Social Sciences Romania Moscow, Russia, June 3-5, 2014.

2. Slide 2 of 26 Corresponding author Dr. IOAN DZITAC, Senior Member of IEEE B. & M.Sc. In Mathematics (1977), Ph.D. in Information Sci. (2002) (Babes-Bolyai University of Cluj-Napoca, RO) Professor of informatics at Aurel Vlaicu University of Arad, RO (tenured since 2009) Senior Researcher at Agora University of Oradea & Director of R&D Agora, RO (2012-2016) Adjunct Professor of the School of Management, University of Chinese Academy of Sciences, China (May 2013-May 2016) Co-founder and General Chair of International Conference on Computers Communications and Control (ICCCC, since 2006) http://univagora.ro/en/icccc2014/ Co-Founder and Associate Editor in Chief of International Journal of Computers Communications & Control (since 2006), In Science Citation Index Expanded (ISI Thomson Reuters, Impact Factor(IF) in JCR2009 = 0.373; JCR2010 = 0.650; JCR2011 = 0.438; JCR2012 = 0.441); A) Automation & Control Systems [Q4, 49 of 59] ; 2B) Computer Science, Information Systems [Q4, 109 of 132]. In Scopus (SJR2012 =0.297): A) Computational Theory and Mathematics [Q4], B) Computer Networks and Communications [Q3], C) Computer Science Applications [Q3]. http://univagora.ro/jour/index.php/ijccc Rector of Agora University (2012-2016) rector@univagora.ro www.univagora.ro Co-Chair of SS03 in ITQM2013 Suzhou, China S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Co-Chair of SS07 in ITQM2014 Moscow, Russia

3. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 3 of 26

4. 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, 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. S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 4 of 26

5. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 5 of 26

6. Preliminaries (1/5) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 6 of 26

7. Preliminaries (2/5) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 7 of 26 D. Tufis, I. Dzitac, L.A. Zadeh, M.J. Manolescu and F.G. Filip at ICCCC 2008, Oradea, Romania

8. Preliminaries (3/5) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 8 of 26

9. Preliminaries (4/5) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 9 of 26

10. Preliminaries (5/5) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 10 of 26

11. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 11 of 26

12. Affine Fuzzy Relations (1/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 12 of 26

13. Affine Fuzzy Relations (2/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 13 of 26

14. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 14 of 26

15. Linear Fuzzy Relations (1/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 15 of 26

16. Linear Fuzzy Relations (2/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 16 of 26

17. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 17 of 26

18. Convex Fuzzy Relations (1/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 18 of 26

19. Convex Fuzzy Relations (2/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 19 of 26

20. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 20 of 26

21. M-Convex Fuzzy Relations (1/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 21 of 26

22. M-Convex Fuzzy Relations (2/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 22 of 26

23. Contents Abstract Abstract Preliminaries Affine Fuzzy Relations Linear Fuzzy Relations Convex Fuzzy Relations M-Convex Fuzzy Relations Conclusions & References S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 23 of 26

24. Conclusions & References (1/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 24 of 26 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.

25. Conclusions & References (2/2) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow Slide 25 of 26

26. Thank you for your attention! Acknowledgments This work was supported in part by research centers: 1)Cercetare Dezvoltare Agora (R&D Agora) of Agora University of Oradea (Director: I. Dzitac) and 2) Mathematical Models and Information Systems, Faculty of Exact Sciences of Aurel Vlaicu University of Arad. (Director: I. Dzitac) S. Nadaban, I. Dzitac, Special Types of Fuzzy Relations, 4rd of June, 2014, Moscow