S-Algebra On Partial Ordered Set And Relations Between S-Algebra And Some Logic Algebras

Non-classical logic is the theortical basis of fuzzy reasoning andfuzzy control. In recent years, the application of fuzzy control has obtained a greetsuccess. But fuzzy reasoning, the core of fuzzy control is suspected because of lackingsolid mathematical foundation. In order to solve this prblem, a new research fieldhas been built up and some scholars devoted themselves to this field. Fuzzy logicis studied with algebraic tools in this paper. A kind of algebraic abstract of fuzzylogic, S-algebra on a partical ordered set is given.The main results of this paper are given as follows:Chapter One: The concept of Basic S-Algebra and S-Algebra on a partialordered set are introduced, then the basic properties are given, the sufficient andnecessary conditions of S-Algebra to be regular are discussed, the independence ofS-Algebra is investigated. All above is the basis of the later chapters.Chapter Two: It is pointed out that S-Algebra defined in this paper is nota lattice. Because some good properties only can be found in an algebra structureif the algebra is a lattice. In this part, some further properties are given. Bymeans of the theory of lattices some lattice theoretic properties of S-Algebra andthe conditions for a S-Algebra being a lattice are found.Chapter Three: Some other conditions of which the S-Algebra on the partialordered set should be satisfied in a logic system are given. The relations betweenS-Algebra and some famous algebra are discussed. In this part, the relations be-tween S-Algebra and MV-algebra, BL-Algebra, Heyting Algebra, BCK-Algebra, R₀-Algebra and Boole-Algebra are given. All these studies will enrich the content ofS-Algebra.