Complete Lattice L Pre-i Topology

The article [1] systemly and deeply studied cotopology on completely distributive lattices . We know that complete lattices is the promotion of completely distributive lattices, pre-cotopology is the promotion of cotopology. So. this paper will continue the work of topological molecular lattices and determination of L-pretopology --mainly study the determination of pre-cotopologies on a complete lattices. The following are the construction and main contents of this paper:In chapter 1, we give the basic concepts and conclusions in collections, lattices theory, topological molecular lattices and pretopological molecular lattices for convenience of the following discussion.In chapter 2, firstly, we give the concepts of outside-derived element for a given element; discuss the nature of closure and the nature of outside-derived element in a pretopological molecular lattices. Secondly, we definite pre-closure operators, pre-outside-derived operators, family of pre-outside-derived operators on a complete lattice L. Similar to topological molecular lattices, we introducue remote neighborhood method and on this basis, we give the concepts and propositions about attachment point and outside-rally point. Finally, we prove that in the definition of the appropriate relationship (?) between the sequence, both (CT(L), (?)) (the set of all pre-cotopologies on L) and (CL(L), (?)) (the set of all pre-closure operators on L) are complete lattices which are isomorphic each other, both (CT(L). (?)) and (FD(L), (?)) (the set of all families of pre-outside-derived operators on L) are complete lattices which are isomorphic each other. In other words, pre-cotopologies can be determined by pre-closure operators or families of pre-outside-derived operators.In chapter 3, for the complete lattices which with an order-reversing involusion, firstly. we introduce concepts of interior on the pretopological molecular lattices and concepts of pre-interior operators on the complete lattices; discuss the nature of interior: prove that (IN(L), (?)) (the set of all pre-interior operators) is complete lattice which is isomorphic with (CT(L).(?)). In other words, pre-cotopologies can be determined by pre-interior operators. Secondly, we introduce the concepts of families of covers on L, analyse the relations between families of covers and pre-interior operators, prove that in the definition of the appropriate relationship (?) between the sequence. (FC(L), (?)) (the set of all families of covers on L) is complete lattice which is isomorphic with (CT(L). (?)). In other words, pre-cotopologies can be determined by families of covers on L. Finally, we introduce concepts of pre-difference derived operators on a special complete lattice. For the complete lattices L = 2~X. we prove that appropriate order relations (?) can be defined on PDD(L) (the set of all pre-difference derived operaters on L). such that (PDD(L). (?)) is complete lattice which is isomorphic to (CT(L),(?)). In other words, pre-cotopologies can be determined by pre-difference derived operators on L.