Minimal primes of ideals arising from conditional independence statements
type: Item
instance of: scholarly article
title: Minimal primes of ideals arising from conditional independence statements
zbMATH Open document ID: 1297.13006
author: Irena Swanson
author: Amelia Taylor
published in: Journal of Algebra
publication date: 2014-03-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5604
review text: Let \(n, r_1, \dots, r_n\) be positive integers and set \(N=[r_1]\times \dots \times [r_n]\). Let \(R\) denote the polynomial ring over a field in variables \(x_a\) as \(a\) varies over \(N\). Let \(L\subseteq [n]\) and \(a, b \in N\). The switch function \(s(L,a,b)\) is an element of \(N\) whose \(i\)th component is \(s(L,a,b)_i=b_i\) if \(i\in L\) and \(s(L,a,b)_i=a_i\) otherwise. If \(L=\{j\}\), we simply write \(s(L,a,b)=s(j,a,b)\). For any two indices \(a, b\) in \(N\) we define the distance between them to be \[ d(a, b)=\# \{i: a_i\neq b_i\}. \] Also, for any \(L\subseteq [n]\) and \(i\in [n]\) we define: \[ f_{L, a, b}=x_ax_b-x_{s(L,a,b)}x_{s(L,b,a)}, \] \[ f_{i, a, b}=x_ax_b-x_{s(i,a,b)}x_{s(i,b,a)}. \] For any \(t\in [n]\), define \[ I^{\langle t \rangle}=\langle f_{i, a, b}: a, b \in N, d(a, b)=2, i\in [t]\rangle, \] \[ \widetilde{I^{\langle t \rangle}}=\langle f_{i, a, b}: a, b \in N, i\in [t]\rangle. \] A subset \(S\) of \(N\) is called \(t\)-switchable if for all \(a, b\in S\) with \(d(a, b)=2\), if \(i\in [t]\), then \(s(i,b,a)\in S\). Moreover, for a \(t\)-switchable subset \(S\) of \(N\), we define \[ \widetilde{I^{\langle t \rangle}}_S=\langle f_{i, a, b}: a, b\,\,\;\text{connected in}\, S \}, \,\,\, \mathrm{Var}^{\langle t \rangle}_S=\langle x_a: a\not\in S\rangle, \,\,\, \text{and}\,\,\, \widetilde{P^{\langle t \rangle}}_S=\mathrm{Var}^{\langle t \rangle}_S+\widetilde{I^{\langle t \rangle}}_S. \] Here two elements \(a, b\in S\) are connected in \(S\) if there is \[ a_0=a, a_1, \dots, a_{k-1}, a_k=b\in S, \] such that for all \(1\leq j\leq k\), \(a_{j-1}\) and \(a_j\) differ only in one component. We say that a \(t\)-switchable subset \(S\) of \(N\) is maximal if for all \(t\)-switchable subset \(T\) of \(N\) properly containing \(S\), \(P^{\langle t \rangle}_S\) and \(P^{\langle t \rangle}_T\) are incomparable. In this interesting paper, the authors by using a new combinatorial structure and \(t\)-switchable subsets, describe of the prime ideals m…
reviewed by: Monireh Sedghi
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 6267891
zbMATH Keywords: primary decomposition
zbMATH Keywords: conditional independence ideals
zbMATH Keywords: graphical modules
zbMATH Keywords: Gröbner bases
zbMATH Keywords: binomial ideals
describes a project that uses: Binomials.m2
describes a project that uses: Macaulay2
MaRDI profile type: Publication
OpenAlex ID: W2086157622
arXiv ID: 1107.5604
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DOI: 10.1016/J.JALGEBRA.2013.05.019
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description: scientific article