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In this article we study the (finite-type, cl-closed) closure operations on ideals of a BCK-algebra, with an emphasis on structural properties. Also we give several theorems that make different closure operations, especially on a quotient and Noetherian BCK-algebra. In particular, we show that the intersection and union (by imposing some additional conditions) of closure operations is a closure operation and every closure operation is a finite type on any finitely generated ideal. Moreover by given the notions of residuated quotient ideals and the meet of two ideals, we conclude some related results.