In this paper we introduce and study two new notions of sets called I-mng-closed sets and mn-Ig-closed sets, which are defined on a nonempty set with two minimal structures and an ideal. These sets enables us to obtain unified properties for certain notions of generalized closed sets with respect to an ideal. Moreover, we establish a new separation property called the class mn-TI ,w hich generalizes the class mn-T1/2 (17).