Fuzzy Filters In Effect Algebras And The Relationship Between Effect Algebras And Two Fuzzy Algebraic Systems

In 1936, J.Von. Neumann and G.Birkhoff proposed the concept of quantum logic in their monograph" the logic of quantum Mechanics ", and many relative papers have been published. One of important methods is proposing some algebraic structure as models for quantum logic.In this point of view, ,F.Kopka introduced the concept of D-poset, which generalized orthoalgebras(orthomodular posets);while other persons such as D.Foulis gave the definitions of effect algebras in another point of view. In addition,orthoalgebra was introduced by R.Giuntion and H.Grealing.The three algebraic structures above generalized quantum logic. This thesis mainly focus on the relative questions in effect algebras and pseudoeffect algebras.Effect algebras contain the usual studying quantum logics,such as orthomod-ular lattice and posets,as its special cases. Their applications are very widely.So the studies in their algebraic properties are very important , especially play a role not to be ignored in the basic study of quantum logics. Ideals and filters are very' important properties in effect algebras. In 1992,Foulis studied the local filters in orthoalgebras. In 2001, Jenca and Pulmannova introduced the definitions of filters in lattice ordered effect algebras and studied their properties. And,Shang Yun and professor Li Yongming gave the definitions of generalized ideals in orthoalge-bras and got some interested properties in 2003. After a time,definitions of ideals and filters in pseudoeffect algebras were introduced by Wu Jing in his paper.Of course,there are many papers about filters and ideals. However,does there ex-ist fuzzy filters or fuzzy ideals in effect algebras?If they exist, what properties they have?Many relative questions do not answer to far. Pseudoeffect algebras are introduced for studying some new coming things. Then,is there relative problems above in pseudoeffect algebras?This thesis tries to answer these questions.Fuzzy logics and quantum logics as well as their algebraic system are very active in non-classical logic now.MV-algebra was introduced by C.Chang in 1958. It is the corresponding algebraic system of Lukasiewicz fuzzy logic systems.And MV-algebras play a very important role in effect algebras .R₀-algebra was introduced by professorWang Guojun.It is the algebraic model of proposition fuzzy logic system ï¿¡*,& very interested system introduced by Wang himself f20'21' .What is the relationship between effect algebras and the two algebraic systems above?This is a question to be worth to study.The present thesis mainly focus on the questions above.The thesis is divided into three parts.The first part is Chapterl.In Chapterl,we investigate their properties by introducing the definitions of fuzzy filters and fuzzy ideals in effect algebras,and get the conclusion that fuzzy filters are a useful tool to obtain the results on classical ideals.We give the definitions of strong fuzzy filters (which is the special cases of fuzzy filters),and prove that the set of all strong fuzzy filters satisfying some certain conditions makes a D-poset under the operation ? we defined.Moreover,we introduce a fuzzy congruence in linearly ordered effect algebras and prove that the fuzzy quotien by this fuzzy congruence relation are also linearly ordered effect algebras..The second part is Chapter2.In this part,we give the definitions of fuzzy filters and fuzzy ideals .Though the concept of right trigle angle and left trigle angle,we investigate the properties of fuzzy filters and fuzzy ideals and get several good conclusions.The third part of this article is Chapter3.In this part,at first,we investigate the questions about compatibility,and obtain some properties about compatibility .In second, we give an implicative operation and get many properties about the implicative operation.Then we get the relationship between effect algebras and MV-algebras as well as .Ro-algebras under the implicative operation we give. We prove that though our implicative operation ,linearly effect algebras are MV-algebras and they are also i?o-algebras. And on the other hand,Ho-algebras are also able to be effect algebras.