2007/07/19 by Ali Soleyman Jahan, Jahan, Ali Soleyman, Xinxian Zheng +1 · 1 citation
Computer Science · Mathematics · #13A30 #13F20 #13F55 #16W70 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0707.2914
openalex publication_date 2007/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Björner 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.