Research On The Algebra Structure Of Soft Set And Its Application

The fuzzy set theory introduced by American Cybernetics expert Zadeh,the rough set theory introduced by Polish mathematician Pawlak in1982and the soft set theory introduced by Russian mathematician Molodtsov have shown their own advantages and characteristic in dealing with uncertainty. The three kinds of theories complement each other and they are closely related.Now the research on the soft set theory focuses on three major aspects:the first is perfecting the soft theory, the second is merging it with the other studied fields; the third is on its application. As three kinds of major mathematical tool dealing with uncertainty, the study merging the fuzzy set theory, the rough set theory and the soft set theory is one of the hot spots at present.It is well known that the urban traffic congestion is a general problem in the development of every country in the world. With the increase of traffic demand, urban traffic congestion becomes more serious. The direct and indirect economic loss caused by traffic congestion is amazing. How to predict accurately the forthcoming congestion and to take measures timely is one of the hot spots in the study on urban traffic congestion.The intent of this paper is studying on the problems merging with the soft set theory,the fuzzy set theory, the rough set theory and the lattice theory and discussing the properties of the related models and the algebraic structures under the related operators. Based on the results, this paper presents a method predicting urban traffic congestion.The main results and innovations in this thesis are summarized as follows:1. The notion of the soft lattice is presented and the properties that the soft latt∩ice is closed under the operators∩ε,∩,∪and∧are derived. By means of∈-soft sets and q-soft sets, some characterizations of idealistic soft lattices and filteristic soft lattices are investigated. Based on the notion of the soft fuzzy set presented by Maji et al., the notions of the soft fuzzy lattice and the soft fuzzy ideal on a lattice are presented and their algebraic structures are discussed. The notion of generated fuzzy sublattices of an arbitrary subset of a lattice is presented and the method getting the generated fuzzy sublattices of an arbitrary subset is presented. The operator U between the fuzzy sublattices is inducted. The results are derived that the set of all fuzzy sublattices of a lattice forms a lattice on the operators∪and∩and that the set of all soft fuzzy sublattices forms a lattice on the operators U and∩.2. Applying the soft set theory to the RSL-algebra,the notion of the soft RSL-algebra is presented and some related properties are derived, and the operators between the soft RSL-subalgebras∪,∩,∪ε and∩ε are defined. The result is derived that the RSL-algebra is isotonic under a homomorphic mapping.3. Based on the notions of the soft groups presented by Aktas and Cagman and the soft rings presented by Acar, their algebraic structures are discussed, respectively. Based on the operator∪between the subgroups of a commutative group, the operator U between the soft groups on a commutative group is inducted and the results are derived that the subgroups of a commutative group and the soft groups on a commutative group are closed under the operators U and U, respectively. The result is derived that the set of all soft groups over a commutative group on the operators∩and∪forms a lattice.4. Applying the soft set theory to the the rough set theory, the model of the soft rough set is constructed and its algebraic structures are discussed. The result is derived that the set of soft rough sets over a given initial universe forms a lattice. A model of soft fuzzy rough set based on a soft set is constructed, and the operators∪,∪ε,∩and∩ε between the soft fuzzy rough sets are presented. The results are derived that the set of soft fuzzy rough sets over a given initial universe under the operators∩and∪ε,∩ε and∪, forms a distributive lattice, respectively.5. Based on the soft fuzzy rough set theory, a method predicting on urban traffic congestion is discussed.