On General Closure Operators and Quasi Factorization Structures

In this article the notions of (quasi weakly hereditary) general closure operator $\mb{C}$ on a category $\cx$ with respect to a class $\cm$ of morphisms, and quasi factorization structures in a category $\cx$ are introduced. It is shown that under certain conditions, if $(\ce, \cm)$ is a quasi factorization structure in $\cx$, then $\cx$ has quasi right $\cm$-factorization structure and quasi left $\ce$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $\cm$, every quasi factorization structure $(\ce, \cm)$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $\cm$, 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 furnished.