L-closure Systems And L-fuzzy Weights

Determining a topology is an interesting question. For a given set X, let T(X) be the set of all topologies on X, and CL(X) the set of all Kuratovski closure operators on X. If one can give both an appropriate order relation≤on CL(X) and an order isomorphism from (CL(X),≤) to (T(X),(?)), then we say that topologies and closure operators can be determined reciprocally. It can be proved that topologies and each of closure operators, interior operators, exterior operators, boundary operators, derived operators, difference derived operators, neighborhood operators, remote neighborhood operators, and convergence classes of nets, can be determined reciprocally. Similar results on L-closure systems will be proved in this paper.The main points of this paper are as follows:In the first chapter, some concepts of L-closure systems and category are intro-duced.In the second chapter, let CS(X,L) be the set of all L-closure systems (where L is a complete De Morgan algebra). In this paper, we give appropriate order relations on WCL(X, L) (the set of all L-weak closure operators of X), WIN(X, L) (the set of all L-weak interior operators of X), WE(X,L) (the set of all L-weak exterior operators of X), we prove that they are complete lattices that are ismorphic to (CS(X, L), (?)). and it gives L-CS (the category of L-closure spaces and continuous mappings) is topological construct.In the third chapter, let X be a set, L a Hutton algebra, FW(X, L)the set of all L-fuzzy weights on X, FN(X, L) the set of all L-fuzzy neighborhood systems on X, In this chapter, we give a one-to-one correspondenceφ12 from FW(X, L) to FN(X,L).