Some Research On Fuzzy Incline Algebras

In this paper, the definitions of (∈,∈∨q(λ,μ))-fuzzy subinclines, (∈,∈∨q(λ,μ))-fuzzy ideal of incline algebras, the new concept of soft subinclines, anti-soft subinclines were given. And some of their equivalent conditions and related properties were discussed.The main results obtained in this dissertation are summarized as follows:In chapter three, we introduce the concept of (∈,∈∨q(λ,μ))-fuzzy subincline and discuss some equivalent characterizations. Then we prove the union of (∈,∈∨q(λ,μ))-fuzzy subinclines is also (∈,∈∨q(λ,μ))-fuzzy subincline. Some relative properties of itshomomorphic image and homomorphicpreimage are obtainedbased on the anti-extension principle. Besides, by introducing anti-direct product of fuzzy sets, we prove the anti-direct product of (∈,∈∨q(λ,μ))-fuzzy subinclines is still (∈,∈∨q(λ,μ))-fuzzy subincline.Finally, we discuss some basic properties of (∈,∈∨q(λ,μ))-fuzzy subinclines throngh the chain condition.In chapter four, firstly, the concept of (∈,∈∨q(λ,μ))-fuzzy ideal of incline algebra is introduced and some equivalent characterizations are discussed. Then the union and the anti-direct product of (∈,∈∨q(λ,μ))-fuzzy ideals of incline algebras are still (∈,∈∨q(λ,μ))-fuzzy ideals of incline algebras is proved. Besides, some relative properties of its isomorphic image and homomorphic preimage are obtained based on the anti-extension principle. Finally, we discuss some basic properties of (∈,∈∨q(λ,μ))-fuzzy ideals of incline algebras throngh the chain condition of filters of incline algebras.In chapter five, firstly, we introduce the new concept of soft subincline. We then obtain some equivalent conditions of it and discuss two operations "RESTRICTED INTERSECT" and "AND" on it. After that we study the relationship between soft subincline and the dual of soft set based on the method of the dual of soft set. In addition, the concepts and properties of maps between soft subincline are given. Finally, we introduce the chain condition of H which consists of all of the soft subinclines and obtain a necessary and sufficient condition for H is Artinian or Noetherian.In chapter six, firstly, the new concept of anti-soft subinclines is given. We then study the relationship between anti-soft subinclines and the anti-dual of soft sets and get some equivalent characterizations. After that we discuss some properties of the two operations ’RESTRICTED UNION’and ’OR’ on it. In addition, based on the anti-extension principle, the properties of their anti-image and inverse image are given. Finally, we introduce the chain condition of H which consists of all of the anti-soft subinclines and obtain a necessary and sufficient condition for H is Artinian or Noetherian.