New methods for constructing shellable simplicial complexes
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...
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Kharazmi University
2022-12-01
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| Series: | پژوهشهای ریاضی |
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| Online Access: | http://mmr.khu.ac.ir/article-1-3171-en.html |
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| author | Mohammad Farrokhi D. G. Ali Akbar Yazdan Pour |
| author_facet | Mohammad Farrokhi D. G. Ali Akbar Yazdan Pour |
| author_sort | Mohammad Farrokhi D. G. |
| collection | DOAJ |
| description | 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<cdots
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| first_indexed | 2024-04-10T00:46:20Z |
| format | Article |
| id | doaj.art-a71a5438c77742a8a3791c323b375a13 |
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| issn | 2588-2546 2588-2554 |
| language | fas |
| last_indexed | 2024-04-10T00:46:20Z |
| publishDate | 2022-12-01 |
| publisher | Kharazmi University |
| record_format | Article |
| series | پژوهشهای ریاضی |
| spelling | doaj.art-a71a5438c77742a8a3791c323b375a132023-03-13T19:26:42ZfasKharazmi Universityپژوهشهای ریاضی2588-25462588-25542022-12-0184164179New methods for constructing shellable simplicial complexesMohammad Farrokhi D. G.0Ali Akbar Yazdan Pour1 Institute for Advanced Studies in Basic Sciences (IASBS) Institute for Advanced Studies in Basic Sciences (IASBS) 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<cdots </cdotshttp://mmr.khu.ac.ir/article-1-3171-en.htmlclutterhybrid cluttershellabilitycohen-macaulayindependence complex |
| spellingShingle | Mohammad Farrokhi D. G. Ali Akbar Yazdan Pour New methods for constructing shellable simplicial complexes پژوهشهای ریاضی clutter hybrid clutter shellability cohen-macaulay independence complex |
| title | New methods for constructing shellable simplicial complexes |
| title_full | New methods for constructing shellable simplicial complexes |
| title_fullStr | New methods for constructing shellable simplicial complexes |
| title_full_unstemmed | New methods for constructing shellable simplicial complexes |
| title_short | New methods for constructing shellable simplicial complexes |
| title_sort | new methods for constructing shellable simplicial complexes |
| topic | clutter hybrid clutter shellability cohen-macaulay independence complex |
| url | http://mmr.khu.ac.ir/article-1-3171-en.html |
| work_keys_str_mv | AT mohammadfarrokhidg newmethodsforconstructingshellablesimplicialcomplexes AT aliakbaryazdanpour newmethodsforconstructingshellablesimplicialcomplexes |