Study On Interval Algebra Of Residuated Lattice Values

The progress of science and technology is closely connected with the development of modern mathematics,especially non-classical mathematical logic,including multivalued logic and fuzzy logic,which plays an irreplaceable role in artificial intelligence and other fields.It lays a theoretical foundation for dealing with uncertain information and automatic reasoning,and studies of logic algebra just a powerful tool for studying non-classical mathematical logic.For example,the completeness of multivalued Lukasiewicz logic system can be proved by MV algebra.Quantum mechanics has become one of the two basic pillars of modern physics,mathematical description of physical phenomena in quantum mechanics is one of the main research direction in the quantum theory.Quantum logic is a propositional calculus associated with the event structure of physical phenomena,it constructs the mathematical framework of quantum mechanics.With the deep research of quantum logic,we directly verify that the event structure set in the quantum theory is an orthomodular lattice through quantum experiments,and orthomodular lattice has been regarded as a main mathematical model for quantum mechanics.Basic algebra is an algebraic structure introduced by Chajda in order to study the generality of MV algebra and orthomodular lattice,it plays a very important role in multivalued logic and quantum mechanics.Up to now,there have been many discussions about the nature and structure of basic algebra.As a generalization of MV algebra,basic algebra has many connections with MV algebra.Every MV algebra is a commutative basic algebra,a finite commutative basic algebra is an MV algebra,but there exists an example of a commutative basic algebra which is not an MV algebra.Further,the basic algebra is MV algebra if and only if it satisfies the associative law.Therefore,many conclusions of MV algebra can be extended to basic algebra.Mundici took the interval of MV algebra as the model,proposed the concept of interval MV algebra,and proved that IMV algebras are categorically equivalent to MV-algebras,but IMV algebra and MV algebra is not term equivalent.Further,he gave a representation of free IMV-algebras.In this thesis,by using IMV’s construction method,we propose interval basic algebra,and study its algebraic structure and related properties.At the same time,we extend sheffer stroke operation to interval basic algebra,and put forward the concept of sheffer stroke interval basic algebra.Some properties of sheffer stroke interval basic algebra are studied.This thesis includes the following contents concretely:1.In this thesis,the definition of interval basic algebra is given,and give an example of the existence of the interval basic algebra.By defining operation ?,? on an interval basic algebra,it becomes a lattice structure,and it is proved that the interval basic algebra satisfying certain conditions is a residuated lattice.2.We extend the sheffer stroke operation to interval basic algebra,put forward the concept of sheffer stroke interval basic algebra,and define the partial order of sheffer stroke interval basic algebra,and its sequence structure and properties are studied.We discuss therelationship between sheffer stroke interval basic algebras and interval basic algebras and the one-to-one correspondence between sheffer stroke interval basic algebras and interval basic algebras satisfying certain conditions is established.3.The relationship between interval basic algebras and interval MV algebras is discussed and the necessary and sufficient condition for interval basic algebras to be interval MV algebras is given.In addition,a construction method of interval basic algebra is presented,the bridge between basic algebra and interval basic algebra is set up.