Seyed Shahin Mousavi Mirkalai - Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Naser Hosseini - Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran
Azadeh Ilaghi-Hosseini - Department of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of Kerman

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

Quasi mono (epi), quasi (right, left) factorization structure, (quasi weakly hereditary, quasi idempotent) general closure operator