On General Closure Operators and Quasi Factorization Structures

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