Paper:

# Fuzzy Multisets in Granular Hierarchical Structures Generated from Free Monoids

## Tetsuya Murai^{*1}, Sadaaki Miyamoto^{*2}, Masahiro Inuiguchi^{*3},

Yasuo Kudo^{*4}, and Seiki Akama^{*5}

^{*1}Graduate School of Information Science and Technologies, Hokkaido University, Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido 060-0814, Japan

^{*2}Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan

^{*3}Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

^{*4}Department of Computer Science and Systems Engineering, Muroran Institute of Technology, 27-1 Miumoto, Muroran, Hokkaido 050-8585, Japan

^{*5}C-Republic, 1-20-1 Higashi-Yurigaoka, Asoh-ku, Kawasaki, Kanagawa 215-0012, Japan

Fuzzy multisets defined by Yager take multisets on interval (0,1] as grades of membership. As Miyamoto later pointed out, the fuzzy multiset operations originally defined by Yager are not compatible with those of fuzzy sets as special cases. Miyamoto proposed different definitions for fuzzy multiset operations. This paper focuses on the two definitions of fuzzy multiset operations, one by Yager and the other by Miyamoto. It examines their differences in the framework of granular hierarchical structures generated from the free monoids as proposed in our previous papers. In order to define basic order between multisets on interval (0,1], Yager uses the natural order on the range **N**, the set of natural numbers, whereas Miyamoto newly introduces an order generated from *both* domain (0,1] and range **N** through the notion of cuts.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.19, No.1, pp. 43-50, 2015.

- [1] W. D. Blizard, “Multiset theory,” Notre Dame J. of Formal Logic Vol.30, pp. 36-66, 1989.
- [2] S. P. Jena, S. K. Ghosh, and B. K. Tripathy, “On the theory of bags and lists,” Information Sciences, Vol.132, pp. 241-254, 2001.
- [3] D. E. Knuth, “The Art of Computer Programming,” Vol.2, Addison-Wesley, Reading, Massachusetts, 1969.
- [4] D. Singh, A. M. Ibrahim, A. Bello, T. Yohanna, and J. N. Singh, “A systematization of fundamentals of multisets,” Lecturas Matemáticas, Vol.29, pp. 33-48, 2008.
- [5] C.-C. Chan, “Learning rules from very large databases using rough multisets,” Trans. on Rough Sets I, LNCS 3100, pp. 59-77, 2004.
- [6] K. P. Girish and J. J. Sunil, “Rough multisets and information multisystems,” Advances in Decision Sciences, Vol.2011, 2011.
- [7] B. Li, “Fuzzy bags and applications,” Fuzzy Sets and Systems, Vol.34, pp. 61-71, 1990.
- [8] S. Miyamoto, “Generalizations of multisets and rough approximations,” Int. J. of Intelligent Systems, Vol.19, pp. 639-652, 2004.
- [9] S. Miyamoto, “Different generalizations of bags,” Annals of Operations Research, Vol.195, pp. 221-236, 2012.
- [10] D. Rochester and P. Bosc, “The set of fuzzy relative integers and fuzzy bags,” Int. J. of Intelligent Systems, Vol.24, pp. 677-694, 2009.
- [11] R. R. Yager, “On the theory of bags,” Int. J. General Systems, Vol.13, pp. 23-37, 1986.
- [12] S.Miyamoto, “Fuzzy multisets and fuzzy clustering of documents,” Proc. of 10th IEEE Int. Conf. on Fuzzy Systems, pp. 1539-1542, 2001.
- [13] A. Syropoulos, “Mathematics of multisets,” C. S. Calude et al. (eds.): Multiset Processing, LNCS 2235, Springer, pp. 347-358, 2001.
- [14] G. Lamperti, M. Melchiori, and M. Zanella, “On multisets in database systems,” Multiset Processing, LNCS 2235, pp. 147-215, 2001.
- [15] Y.-K. Chen and H.-C. Liao, “An investigation on selection of simplified aggregate production planning strategies using MADM approaches,” Int. J. of Production Research, Vol.41, pp. 3359-3374, 2003.
- [16] A. Rebaï and J.-M. Martel, “A fuzzy bag approach to choosing the “best” multiattributed potential actions in a multiple judgement and non cardinal data context,” Fuzzy Sets and Systems, Vol.87, pp. 159-166, 1997.
- [17] T. Murai, S. Miyamoto, M. Inuiguchi, and S. Akama, “Granular Hierarchical Structures of Finite Naïve Subsets and Multisets,” Int. J. of Reasoning-based Intelligent Systems, Vol.4, No.3, pp. 118-128, 2012.
- [18] T. Murai, S. Miyamoto, M. Inuiguchi, Y. Kudo, and S. Akama, “Crisp and Fuzzy Granular Hierarchical Structures Generated from a Free Monoid,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.18, No.6, pp. 929-936, 2014.
- [19] L. A. Zadeh, “Fuzzy sets,” Information and Control Vol.8, No.3, pp. 338-353, 1965.
- [20] T. Tanaka, T. Murai, Y. Kudo, and S. Akama, “Rough sets in crisp and fuzzy granular hierarchical structures,” Proc. IEEE GrC2014, to appear, 2014.
- [21] Z. Pawlak, “Rough sets: Theoretical aspects of reasoning about data,” Kluwer, Dordrecht, 1991.