Study some Modern Topological and Algebraic Structures

Abstract

Multiset theory was introduced in 1986 by Yager [68]. A multiset is considered to be the generalization of a classical set. In classical set theory, a set is a wellde ned collection of distinct objects. It states that a given element can appear only once in a set without repetition. So, the only possible relation between two mathematical objects is either they are equal or they are di erent. The situation in science and in ordinary life is not like this. If the repetitions of any object is allowed in a set, then a mathematical structure, that is known as multiset (mset [9] or bag [68], for short), is obtained in [11, 25, 59, 60]. For the sake of convenience an mset is written as fk1=x1; k2=x2; :::; kn=xng in which the element xi occurs ki times. The number of occurrences of an object x in an mset A, which is nite in most of the studies that involve msets, is called its multiplicity or characteristic value, usually denoted by mA(x) or CA(x) or simply by A(x). Noted that each multiplicity ki is a positive integer. In Mathematics, the equation x2􀀀4x+4 = 0 has a solution x = 2; 2 which gives the multiset S = f2=2g. Additionally, One of the simplest examples is the multiset of prime factors of a positive integer n. The number 504 has the factorization 504 = 233271 which gives the mset X = f3=2; 2=3; 1=7g where CX(2) = 3, CX(3) = 2, CX(7) = 1. In Chemistry, a water molecule H2O is represented by the mset M = f2=H; 1=Og and without one of the two hydrogen atoms, the water molecule is not created. In the physical world it is observed that there is enormous repetition. For instance, there are many hydrogen atoms, many water molecules, many strands of DNA, etc. This leads to three possible relations between any two physical objects; they are di erent, they are the same but separate or they coincide and are identical. Many conclusive results were established by these authors and further study was carried on by Jena et al. [32] and many others [12, 13, 14, 47]. The notion of an mset is well established both in mathematics and computer science [9, 10, 15, 16, 23, 58, 61, 62]. iii A wide application of msets can be found in various branches of mathematics. Algebraic structures for mset space have been constructed by Ibrahim et al. [29]. In [53], Okamoto et al. used msets in coloring of graphs. Additionally, application of mset theory in decision making can be seen in [69]. In 2012, Girish and Sunil [26] introduced multiset topologies induced by mset relations. The same authors in [27] further studied the notions of open sets, closed sets, basis, sub-basis, closure, interior, continuity in multiset topological (M-topological, for short) spaces. The concept of soft sets was rst introduced by Molodtsov [48] in 1999 as a general mathematical tool for dealing with uncertain objects. In [48, 49], Molodtsov successfully applied the soft set theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory of measurement, and so on. In 2011, Shabir and Naz [56] initiated the study of soft topological spaces. They de ned soft topology on the collection of soft sets over X. Consequently, they de ned basic notions of soft topological spaces such as open soft sets and closed soft sets, soft subspace, soft closure, soft nbd of a point, soft separation axioms, soft regular spaces and soft normal spaces. In 2013, Babitha et al. [8] and Tokat et al. [64] introduced the concept of soft mset (F;E) as F : E ! PW(U) where E is a set of parameters and PW(U) is a power whole mset of an mset U. Moreover, Tokat et al. [65] introduced the concept of soft mset (F;E) by another way as F : E ! P (U) where E is a set of parameters and P (U) is a power set of an mset U. The notion of a soft multiset in this thesis is the same as in [65, 66, 67]. In 2013, Tokat et al. [64] introduced the concept of soft multi topology and its basic properties. In addition, the notion of soft multi connectedness was studied in [65]. Additionally, the notion of soft multi compactness on soft multi topological spaces was presented in [66]. In 2015, Tokat et al. [67] presented the notion of soft multi continuous functions. The concept of soft msets which is combining soft sets and msets can be used to solve some real life problems. Also, this concept can be used in many areas, such as