Study On The Alpha-Subsets And Inequalities Of Lattice Implication Algebras

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 result of lattice implication algebra, properties and structure are further discussed in this paper. Firstly, we will discuss the li-ideals of finite lattice implication algebra; Secondly, we will continue to discuss the properties of annihilators; Thirdly, we propose the concept of theα—subset in lattice implication algebra and discuss many properties; Lastly, we propose the lattice implication algebric inequality in lattice implication algebra and study the properties of the solution sets. The main results of this paper are listed as follows.1. We will discuss the li-ideals of lattice implication product algebra. The structure of li-ideals finite lattice implication algebra is discussed and a method of finding out all li-ideals of finite lattice implication algebra is presented. The relation between lattice of li-ideals and boolean lattice is proved to be an isomorphic relation in the finite lattice implication algebra.2. We will further discuss the properties of annihilator in lattice implication algebra. As for the li-ideal of finite lattice implication algebra, the doing annihilator is an order-reversing involutive operator. We can define an implicative operator " => "in the setΣ(L) of all li-ideals of L, so (Σ(L),O,L, 0~* ,=> ) is a lattice implication algebra.3. The conpet ofα- subset is proposed in the lattice implication algebra and some properties of theα- subset are obtained in the normal sets. In the second, we study the relationship betweenα- subsets and ideals in the lattice implication algebra. And we will prove that theα- subset of lattice implication algebra is a lattice ideal. In the end, we will prove that the homomorphism image of anα- subset of B is a subset of the f(α) - subset of f(B) in the lattice implication algebra.4. We will propose the concept of lattice implication algebric inequality in the lattice implication algebra. Then we will discuss the solution of three kinds important lattice implication algebric inequality and study the properties of the solution of the inequality in the lattice implication algebra.