Study On The Lattice Implication Algebras And Correlative Substructures

Lattice-valued logic system provides a logic foundation for intelligent information processing. Lattice implication algebra is an important logical algebra, and it offers a theoretical basis for lattice-valued logic and approximate reasoning. Based on the existing results of lattice implication algebra, properties and sub-structure are further discussed in this paper in order to enrich the contents of lattice-valued system and approximate reasoning. The main results of this paper are listed as follows:1. The purpose of this part is to study the sub-structure of lattice implication algebras. An equivalent definition of LI-ideal is proposed. Some properties of lattice implication subalgebra and LI-ideal are proposed. The relation between lattice implication subalgebra and LI-ideal are presented. It is proved that no LI-ideals is non-trivial lattice implication subalgebras.2. The concept of Boolean filters is proposed and its basic properties are discussed, and an important conclusion on the quotient algebras induced by Boolean filters which are Boolean algebras is given. Then most of studies on the relationships among Boolean filters, prime filters, ultra filters and obstinate filters are made. Besides, the fuzzification of the notion of Boolean filters is considered, and the properties of fuzzy Boolean filters are investigated, the equivalence relation between the fuzzy implicative filters and fuzzy Boolean filters is get.3. The structure and properties of subalgebras of direct product of lattice implication algebras are investigated further, and the properties of filters of direct product of two finite chain-type lattice implication algebras are discussed. It is proved that the filters of direct product of two finite chain-type lattice implication algebras have trivial filters and prime filters only. Finaly, the concrete forms of the subalgebras, filters and LI-ideals of direct product of two finite chain-type lattice implication algebras are obtained by means of MATLAB, and it is also necessary to know the structure of lattice implication subalgebras more intuitively.