Ideal theory of pre-logics based on the theory of falling shadows

Based on the theory of a falling shadow which was first formulated by Wang [8], a theoretical approach of the ideal structure in pre-logics is established. The notions of a falling subalgebra, a falling and a positive implicative falling ideal of a pre-logic are introduced. Some fundamental properties are investigated. Relations among a falling subalgebra, a falling ideal and a positive implicative falling ideal are stated. Characterizations of falling deals and positive implicative falling ideals are discussed.