Study Of The Properties Of The Lattice Implication Algebra

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. On the prime ideal space of distributive lattice with bounds, a representation of lattice implication algebra was given. And with the help of the limit of set, the properties of this representation were discussed. Finally, some of properties of any subset's limit of lattice implication algebra have been proved.2.The structure of the lattice implication sub-algebra which generated from a subset, and the product of lattice implication algebra were studied. The necessary condition was given of a lattice implication algebra which can be decomposed into lattice implication algebra's product. It is proved that lattice H implication algebra is a Boole ring with a unit.3.The concepts of O-ideal was proposed and its property was discussed. Ultra-filter on a set S of lattice implication algebra was defined and was compared with the ultra-filter on a set of lattice, the relationship between this two ultra-filter was studied.