Annals of Communications in Mathematics

AbstractThe concept of multisets emerged by violating a principle of distinct collection of object in crisp sets. In some practical situations, multisets with multiplicity of their objects varying overtime are frequently encountered, such multisets are called dynamic multisets. However, there has been no formal mathematical study on dynamic multisets. Dynamic multiset is a special kind of multiset with time varying multiplicity of elements. The importance of dynamic multisets stems from their potential usefulness in resolving a task of finding duplicate records within large databases. In this paper, we vividly explore the concept of dynamic multisets and present some of its properties. We observe that, the operations on dynamic multisets are the same as that of static multisets, with the time parameter as the only distinction. Finally, some application-driven examples of dynamic multisets are presented.

AbstractFuzzy multigroup is a structure that generalizes the idea of fuzzy group. In fact, the concept of fuzzy multigroups is the application of fuzzy multisets to group theory. The idea of direct product of fuzzy multigroups has been established. This paper extends the notion of direct product between two fuzzy multigroups to the case of finitely many fuzzy multigroups. Some properties of generalized direct product of fuzzy multigroups are elucidated. It is shown that generalized direct product of fuzzy multigroups is a fuzzy multigroup. Finally, a number of results are obtained and duly verified with respect to alph-cuts and level sets.

AbstractThis paper is an attempt to summarize the basic concepts of the theories of multiset and fuzzy multiset. We begin by describing multisets and the operations between them with some related results. In the same vein, the basic concepts of fuzzy multiset theory as well as the operations between fuzzy multisets are buttressed in relation to multiset theory. Finally, we present some properties of fuzzy multisets with some related results.