This chapter explores the various algebraic structures that exist in the fuzzy set environment in a multiset context. The authors have also established analogs of groups, rings, and ideals of sets in multiset environment. The conditions under which a multiset that is derived from a group becomes a multiset group (shortly mset group) and that derived from a ring to become a multiset ring (shortly mset ring) have been mentioned. The distinguishing between multiset subring and multiset ideals has been discussed on in the chapter. The authors have also delved into properties such as intersection and union of mset groups, mset rings, and ideals.
Top1. Introduction
Fuzzy Sets have been along for quite a while now. The concept was first introduced by Zadeh in 1965 and since then, a wide variety of applications and developments have been explored. On the other hand, Multisets were introduced almost two decades after fuzzy sets by Knuth, yet the level of advancement is quite pale in comparison. A comprehensive account of fundamentals of multiset and its applications in various forms can be found in (Chris Brink, 1988), C.S.calude et al. (C. S. Calude, G. Paun, G. Rozenberg & A. Salomaa, 2001), D. Singh et al. (D. Singh, A. M. Ibrahim, T. Yohanna & J. N. Singh, 2007), (K. P. Girish & Sunil Jacob John, 2009), (K. P. Girish & S.J. John 2011), (K. P. Girish & S.J. John, 2012), (K. P. Girish & S.J. John, 2012). (N.J.Wildberger, 2003). and (Wayne D. Blizard, 1991). Applications of cardinality bounded multisets can be found in these papers.
Since Crisp Sets have a membership value of 0 and 1, Fuzzy Sets take on a value in [0,1]. But, from this, when we come on to the case of multisets, any integer value both positive and negative is possible. The results of Fuzzy Set Theory have a wide range of application, especially in Computer Science field. Multiset is also now emerging in various fields although it is largely unexplored. A prime example would be GUAVA a version of JAVA developed by Google. It is primarily based on Multisets and their properties rather than Fuzzy Sets.
Multisets are used more and more in computer science for quantitative analysis, information retrieval and models of resources. The concept of multiset was introduced in order to capture the idea of multiplicity of appearance, or resource. Multisets are defined by assuming that for a given set M, an element x occurs a finite number of times. For example, the prime factorization of a natural number ‘n’ is a multiset whose elements are primes. The invariants of a finite abelian group also form a multiset. Even processes in an operating system can be seen as a multiset, and the examples can continue.
Various algebraic structures like Groups, Rings etc. based on Fuzzy Sets are already being used in many fields such as Computer Science, Chemistry, Physics and other subjects. These structures on Fuzzy Sets have been deliberated on for quite a while now. Some of the basic concepts of Fuzzy Groups and Fuzzy Groupoids are explained by Azriel Rosenfeld (Azriel Rosenfeld, 1971). Fuzzy rings, Fuzzy left and right ideals, fuzzy prime and maximal ideals of a general ring R are discussed by Souriar Sebastian et al. (Souriar Sebastian, Mercy K. Jacob, V. M. Mary & Divya Mary Daise. S, 2012). Yuying Li et al. (Yuying Li, Xuzhu Wang & Liqiong Yang, 2013) have presented a special type of (𝜆, 𝜇) Fuzzy subgroup and its basic properties. Even if these exist in multisets, their possibilities and applications have remained largely unexplored. A few like multi-groups and some of their results have been discussed by (A. M. Ibrahim & P.A. Ajegwa 2016), Binod Chandra Ttripathy et al. (Binod ChandraTtripathy, Shyamal Debnath & Debjani Rakshit, 2018), (Johnson Aderemi Awolola & Adeku Musa Ibrahim 2016), (P.A. Ejegwa, 2017), S.K..Nazmul et al. (S.K..Nazmul, P.Majumdar & S.K.Samanta, 2013) and (Y. Tella & S. Daniel, 2013). Different aspects and applications of Fuzzy Multisets are investigated by P. Rajarajeswari and N. Uma (P. Rajarajeswari & N. Uma, 2013) and Sabu Sebastian and T.V. Ramakrishnan (Sabu Sebastian & T.V. Ramakrishnan, 2011).
The authors attempt in this article is to extend the algebraic concepts that are utilized in the Fuzzy environment to a Multiset context. It was noted that while Arrays had an ordered structure, yet allowed for duplicity, and sets had uniqueness in its elements while being unordered, multisets combined the better aspects of the two while removing the drawbacks as the elements of a multiset are both ordered and unique. This observation led to the thought that the usage of multisets and structures derived from them could be used in place of Cantor sets in various places for an improved result, which is what resulted in this paper.
In section 2, some of the definitions of both fuzzy sets and of multisets and few basic concepts about multisets are mentioned. Section 3 deals with group structure, while section 4, ring structure. Meanwhile, the ideals under multiset context are discussed in section 5.