On the relation between filters and deductive rules

Filter is an algebraic structure, which has been widely applied to many branches of mathematics, especially to mathematical logic. Lattice implication algebra is a new kind of logical algebra proposed by Xu Yang. The concepts of filter, implicative filter, positive implicative filter, I-filter, involution filter, obstinate filter and ultra-filter in a lattice implication algebra were proposed and studied. By using the concept of truth functions, this paper is devoted to the study of the relation between filters and logical deductive rules. Further, the concept of G-filter based on deductive rules was proposed with its properties being discussed. The results of this paper provide the new methods for resolution based on filters.