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Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only if $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$,...

For each squarefree monomial ideal $I\subset S=k[x_1,...,x_n]$, we associate a simple finite graph $G_I$ by using the first linear syzygies of $I$. The nodes of $G_I$ are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x,y$ such that $x{u}_i=y{u}_j$. In the cases, where $G_I$ is a cycle or a tree, we show that $I$ has a linear resolution if and only if $I$ has linear quotients and if and only if $I$ is variable-decomposable. In addition, with the same assumption on $G_I$, we characterize all squarefree monomial ideals with a...

Let $K$ be a field and $S=K[x_1,...,x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,...,u_r$ be monomials in $S$. We prove that if $u_1,...,u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,...,u_r)$ is pretty clean. Also, we show that if $u_1,...,u_r$ form a filter-regular sequence on $S/I$, then Stanley’s conjecture is true for $S/I$ if and only if it is true for $S/(I,u_1,...,u_r)$. Finally, we prove that if $u_1,...,u_r$ is a minimal set of generators for $I$ which form either a $d$-sequence, proper sequence or strong $s$-sequence (with respect to the reverse lexicographic...