On Lattice Implication Algebra And The Relation Between It And Correlative Logic Algebras

Lattice implication algebra is an important logical alebra, and it offers a theoretical basis for lattice-valued logic and uncertainty reasoning. Further studies of lattice implication algebra are made in this paper based on the existing results. Firstly, the properties of lattice implication algebra are discussed; Secondly, the properties of filters in lattice implication algebra are discussed, and the concept of submaximal filter is proposed; Lastly, the relations of lattice implication algebra, Wajsberg algebra, fuzzy implication algebra and Ro algebra are discussed. The main results of this paper are listed as follows:1. Some properties of lattice implication algebra are obtained; A distributive quasi-lattice implication algebra is a lattice implication algebra if and only if it satisfies x∨y= (xâ†'y)â†'y. Some properties of operators "(?) "and "(?)" in lattice implication algebra are obtained.2. Some properties of filters in lattice implication algebra are obtained. The relations of filters are discussed. A filter is an ultra-filter if the filter is an implicative filter and a prime filter. The ultra-filter and the obstinate filter are coincident in lattice implication algebra. The concept of fixed point is proposed in lattice implication algebra. There exists no ultra filter in a lattice implication algebra which has a fixed point or has odd elements. Some properties of filter lattice are obtained in lattice implication algebra. The concept of submaximal filter is proposed in lattice implication algebra and related properties of it are obtained. A submaximal filter is a prime filter. The existence of submaximal filter in lattice implication algebra is discussed.3. The relations of lattice implication algebra, Wajsberg algebra, fuzzy implication algebra and Ro algebra are discussed. The condition for a Wajsberg algebra to be transformed to a lattice implication algebra is obtained. The condition for a fuzzy implication algebra to be transformed a lattice implication algebra is obtained. The relation of Ro algebra and lattice implication algebra is discussed.