Closure Systems And Closure Operators On Finite Lattice

In this paper, we introduce new closure systems and closure operators in finite lat-tice, and discuss the relationship between them. we obtain the representation of finitelattices by closure system. Moreover, we discuss the relationship between the corre-sponding lattice category and the corresponding closure system category by takingthese mathematical frameworks as the object and these homomorphic mappings ashomomorphism.Firstly, in section 2, based on the theory of the representation of finite lattices byclosure system, we define a new closure system–distributive closure system, which isused to represent finite distributive lattice. We also find some useful properties andtheorems in distributive closure system. Meantime, we prove that the category formedby closure systems together with homomorphism is equivalent to the category formedby finite lattice together with homomorphism.Secondly, in section 3, we discuss another closure system–atomistic closure system,and get the relationship which is one to one between atomistic closure system andatomistic closure operator, we also study the atomistic implicational system.Finally, we prove that the category formed by atomistic closure systems togetherwith homomorphism is equivalent to the category formed by finite atomistic latticetogether with homomorphism.