Characterizations of MV-Algebras Based on the Theory of Falling Shadows

Based on the falling shadow theory, the concept of falling fuzzy (implicative) ideals as a generalization of that of a T ∧-fuzzy (implicative) ideal is proposed in MV-algebras. The relationships between falling fuzzy (implicative) ideals and T-fuzzy (implicative) ideals are discussed, and conditions for a falling fuzzy (implicative) ideal to be a T ∧-fuzzy (implicative) ideal are provided. Some characterizations of falling fuzzy (implicative) ideals are presented by studying proprieties of them. The product ⊛ and the up product ⊚ operations on falling shadows and the upset of a falling shadow are established, by which T-fuzzy ideals are investigated based on probability spaces.