A multiset approach to algebraic structures sequences and applications

Abstract

As given by George Cantor, the formal definition of set is as a well defined newlinecollection of distinct objects . The terms well defined and distinct in this, and their newlineintegrity, or lack of thereafter is what eventually led to the formation of generalized newlineset theory, mainly consisting of fuzzy sets, multisets, rough sets, soft sets etc. Out of newlinethese, multiset is the primary focus here. The distinctness property is violated in newlinemultisets such that duplicate elements can occur in it. Since multiplicity of elements newlineis the key feature of a multiset, all the findings are based on the count value. In other newlinewords, the characteristic function plays a major role in this work. newlineThe algebraic structures of group, ring, ideal and module have been examined newlineand evaluated in a multiset context. Multiset group, multiset ring, multiset ideal and newlinemultiset module have been defined and some of their properties have been identified newlineand explained. The various features and aspects of these structures within classical newlinesets have been re-evaluated in a multiset point of view. Cosets, factor ring and newlinehomomorphism are some among this. Suitable operations are defined to form the newlineset of all multiset groups as a group and set of all multiset rings as a ring. Different newlinetypes of ideals, such as prime, maximal and principal are considered in multiset newlineenvironment. The multiset group structure is extended to (A,B)- multiset group, newlinewhere A and B are real numbers with A lt B. newlineSequences whose elements are multisets, and their convergence have been studied. newlineA metric is defined on multisets and under this metric, statistical, Wijsman and newlineHausdorff convergences have been studied specifically. Some multiset sequences newlinehave been constructed that have a potential scope of application across different newlineareas. newline