Pretty clean monomial ideals and linear quotients

Abstract

We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal $I$ has linear quotients, then the squarefree part of $I$ and each component of $I$ as well as $\mm I$ have linear quotients, where $\mm$ is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Bj\"orner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly. As a generalization of the facet ideal of a forest, we define monomial ideals of forest type and show that they are pretty clean. This result recovers a recent result of Tuly and Villarreal about the shellability of a clutter with the free vertex property. As another consequence of this result we show that if $I$ is a monomial ideal of forest type, then Stanley's conjecture on Stanley decomposition holds for $S/I$. We also show that a clutter is totally balanced if and only if it has the free vertex property.