Volume 8, Issue 4 (Vol. 8,No. 4, 2022)                   mmr 2022, 8(4): 164-179 | Back to browse issues page

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Farrokhi D. G. M, Yazdan Pour A A. New methods for constructing shellable simplicial complexes. mmr 2022; 8 (4) :164-179
URL: http://mmr.khu.ac.ir/article-1-3171-en.html
1- Institute for Advanced Studies in Basic Sciences (IASBS)
2- Institute for Advanced Studies in Basic Sciences (IASBS) , yazdan@iasbs.ac.ir
Abstract:   (664 Views)
A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}$ we correspond its circuit ideal $I(mathcal{C})$ generated by monomials $x_{i_1}cdots x_{i_k}$ with ${i_1,ldots,i_k}inmathcal{C}$. Conversely, to each square-free monomial ideal $I$ with minimal set of generators $mathcal{G}(I)$, we correspond a clutter with circuits ${i_1,ldots,i_k}$, where $x_{i_1}cdots x_{i_k}inmathcal{G}(I)$. The independence complex of a clutter $mathcal{C}$ on $[n]$ is the simplicial complex $Delta_{mathcal{C}}$ whose faces are independent sets in $mathcal{C}$ by which we mean sets $Fsubseteq [n]$ such that $ensubseteq F$ for all $einmathcal{C}$. It is easy to see that the Stanley-Reisner ideal of $Delta_{mathcal{C}}$ coincides with $I(mathcal{C})$. The above correspondence establishes a one-to-one correspondence between simplicial complexes and independence complex of clutters. A simplicial complex $Delta$ is shellable if there exists a total order on its facets, say $F_1
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Type of Study: S | Subject: alg
Received: 2021/02/15 | Revised: 2023/06/17 | Accepted: 2021/05/1 | Published: 2022/12/31 | ePublished: 2022/12/31

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