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.

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