on general closure operators and quasi factorization structures

In this article the notions of quasi mono (epi) as a generalization of mono (epi), (quasi weakly hereditary) general closure operator c on a category x with respect to a class m of morphisms, and quasi factorization structures in a category x are introduced. it is shown that under certain conditions, if (ε,m) is a quasi factorization structure in x, then x has a quasi right m-factorization structure and a quasi left ε-factorization structure. it is also shown that for a quasi weakly hereditary and quasi idempotent qcd-closure operator with respect to a certain class m, every quasi factorization structure (ε,m) yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class m, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. several illustrative examples are provided.