On Lattice Implication Algebra And Its Especial Filters

The lattice-valued logic extends the chain type true value domain of the multi-valued logic on a relatively general lattice, so that not only the total order information but also the incomparable infonnation can be processed, further the reasoning, judgment and decision of people in the uncertainty environment can be described more effectively, and especially due to the researching on the incomplete comparability of true value, the thinking activity of people can be described more truly. Lattice implication algebra is a kind of algebra structure which combines the lattice and the implication algebra together, and is the foundation of the researching on lattice-value logic system and uncertainty reasoning. Based on the known researching on lattice implication algebra and its substructure, the article further study the properties of the substructure of lattice implication algebra, and mainly consists of the following four parts:1. The concept of n-fold associative filter of lattice implication algebra is introduced. The properties of filters and the differences of different filters in lattice implication algebra are further researched, the mutual transformation conditions of the different filters are provided, and the relations and the mutual transformation conditions of filters are shown in a figure2.2.1and a figure2.2.2.2. The substructure of lattice implication algebra is researched further, the concept of the (∈,∈∨q) fuzzy subalgebra of lattice implication algebra is introduced and studied thoroughly so as to obtain its properties, and finally, the condition that the product of the (∈,∈∨q) fuzzy subalgebra of lattice implication algebra is still the (∈,∈∨q) fuzzy subalgebra is proven.3. The properties of fuzzy filters of lattice implication algebra are researched further. The concepts of fuzzy prime filter, the (∈,∈∨q) fuzzy Boolean filter, the (∈,∈∨q) fuzzy implicative filter, the (∈,∈∨q) fuzzy primer filter are introduced, and their properties are researched thoroughly so as to prove that the (∈,∈∨q) fuzzy Boolean filters and the (∈,∈∨q) fuzzy implicative filters are equivalent.4. The concept of (γ,δ]-fuzzy filter with thresholds in lattice implication algebra is introduced, and the relations between the (γ,δ]-fuzzy filters and level filters of the (γ, δ]-fuzzy filters are discussed so as to obtain some important properties, and provide the equivalent description of the (γ,δ]-fuzzy filters.