Research On Structure Of Lattice Implication Algebras And L-fuzzy Bi-topological Topological Spaces

This academic degree thesis is constituted by two parts:1) Construct theories of Lattice implication algebra; 2) Represent and structure theories of L- fuzzy bi-topological spaces.1. Non-classical logic is an active research direction in the field of artificial intelligence and a logic foundation for uncertainty reasoning. In the study of non-classical logic, lattice-valued logic system is of extensive significance. Lattice implication algebra is an algebraic system combinating lattice with implication algebra. It is first defined by Professor Xu Yang in order to study lattice-valued logic. There are many research papers about lattice implication algebra and related logic. In this paper, the proper and structure of lattice implication algebra is further studied. The following works have been done:1) Research on lattice implication algebra axioms simplification problem. The definition of lattice implication algebra is very trivial; there are more than ten conditions, so there are a lot of literature which has discussed the issue of simplifying the definition of lattice implication algebra. The result on papers [7, 19] is the best. It used to describe the lattice implication algebra by seven conditions. Our justice to lattice implication algebra system does a further simplification. The substance that got lattice implication algebra is a to satisfy four conditions (A1), (A2), (A3), (A4) of (2,0) type algebra. So it is more simplified than the former result consumedly.2) Study the problem of conditions for Chain to be lattice implication algebra. Mainly the following three problems has been solved:(1). Example of the chain that cannot become the standard to contain the algebra . (2). The necessary and sufficient conditions of the chain that may become the standard to contain the algebra .(3). The necessary and sufficient conditions of chain "lattice implication" is the sole operator .3) Study on the algebraic structure of the set of lattice implication algebras. Mainly following three problems has been solved:(1). The order relations on the set of lattice implication algebras in lattice( L ,≤)were introduced.(2). Proved that lattice of lattice implication algebras in finited chain is C2.(3). The uniqueness condition of the lattice implication operators is studied.2. In the L-Fuzzy topological space1) Proved that the L-topological space and the closure of some operations to the internal characteristics. They also discussed the complex nature and the composite results of the quantitative characteristics, a series of interesting results has been instructed.2) Proved that under certain conditions, the compactness in L-fuzzy topological space ( LX ,δ∨) is equivalent to the bi-compactness in L-fuzzy bi-topological spaces ( LX ,δ1 ,δ2).