Some Properties Of Several Types Of Operators On The Poset

This paper takes operators as the main research object,introduces the concepts of irreducible internal operators,irreducible closure operators,Lk-prime closure operators and Lk-prime internal operators,and discusses the related proper-ties of several types of operators.The general structures are as following:Chapter one:The research background and current situation related to this paper are introduced,and we give some basic definitions and theorems that this paper will use.Chapter two:Firstly,the definition of the consistently co-directed complete posets is given,and the properties of projective operators are studied about the con-sistently co-directed complete posets,and the property characterizations of internal operators and closure operators are given.Then,the properties of projective operators are studied on the co-directional complete posets,and the equivalent characterizations of internal operators on the co-directional complete posets are given.Chapter three:Firstly,we combine the definitions of prime closure operator and prime interior operator on the lattice to study the relationships between them and the adjoint.Then,we give the definitions of irreducible interior operator and irreducible closure operator by means of the definitions of meet reducible element and join reducible element on the lattice,and discuss the equivalent characterizations of prime interior operator and irreducible interior operator in the distributive lattice,as well as prime closure operator and irreducible closure operator.Chapter four:With the help of definitions of L-partially ordered sets on the complete residue lattices,the definitions of Lk-prime closure operators and Lk-prime internal operators are given,and their related properties and corresponding equivalent characterizations are studied.On this basis,the definitions and equivalent character-izations of n-multiple Lk-prime closure operators and n-multiple Lk-prime internal operators are generalized.Chapter five:We summarize the results discussed in this paper,and further prospect the issues that may be studied in the future.