On general closure operators and quasi factorization structures

In this article the notions of quasi mono (epi) as a generalization (epi), (quasi weakly hereditary) general closure operator $mathbf{C}$ on category $mathcal{X}$ with respect to class $mathcal{M}$ morphisms, and factorization structures in are introduced. It is shown that under certain conditions, if $(mathcal{E}, mathcal{M})$ structure $mathcal{X}$, then has right $mathcal{M}$-factorization left $mathcal{E}$-factorization structure. also for hereditary idempotent QCD-closure $mathcal{M}$, every yields relative given operator; pair classes dense closed morphisms forms structure, both idempotent. Several illustrative examples provided.