$k$-clean monomial ideals

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show that a $(d-1)$-dimensional simplicial complex is $k$-decomposable if and only if its Stanley-Reisner ideal is $k$-clean, where $k\le d-1$. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are $k$-clean for all $k\ge 0$.