Primary decomposition of ideals of lattice homomorphisms
type: Item
instance of: scholarly article
title: Primary decomposition of ideals of lattice homomorphisms
zbMATH Open document ID: 1395.13007
author: Leila Sharifan
author: Ghazaleh Malekbala
author: Ali Akbar Estaji
published in: The Electronic Journal of Combinatorics
publication date: 2018-08-15T00:00:00Z
full work available at URL: v25i3p8
review text: Given two finite posets \(P\) and \(Q\), a map \(\phi : P \rightarrow Q\) is called isotone if it is order preserving. The set of isotone maps \(P \rightarrow Q\) is denoted by \(\text{Hom}_{\text{Pos}}(P,Q)\). Given two finite lattices \(L\) and \(M\), a map \(\phi : L \rightarrow M\) is called a lattice homomorphism if for any \(l_1, l_2 \in L\), \(\phi(l_1\vee l_2) = \phi(l_1)\vee \phi(l_2)\) and \(\phi(l_1\wedge l_2) = \phi(l_1)\wedge \phi(l_2)\). The set of lattice homomorphisms \(L \rightarrow M\) is denoted by \(\text{Hom}_{\text{Lat}}(L,M)\). Let \(S\) be the polynomial ring over a field \(k\) with variables \(x_{l,m}\) where \(l\in L\) and \(m\in M\). For any \(\phi \in \text{Hom}_{\text{Pos}}(P,Q)\), set \(u_{\phi}=\prod_{l\in L}x_{l,\phi(l)}\). The ideal of poset homomorphisms associated to \(L\) and \(M\) was introduced in [\textit{G. Floystad} et al., J. Pure Appl. Algebra. 221, No. 5, 1218--1241 (2017; Zbl 1356.13023)] and it is defined as \(I(L,M) = (u_{\phi} ; \phi \in \text{Hom}_{\text{Pos}}(P,Q))\). In the paper under review, the authors introduce the ideal of lattice homomorphisms as \(J(L, M) = (u_{\phi} ; \phi \in \text{Hom}_{\text{Lat}}(P,Q))\). It is show that \(L\) is a distributive lattice if and only if the equidimensinal part of \(J(L, M)\) is the same as the equidimensional part of the ideal of poset homomorphisms \(I(L, M)\). The authors study the minimal primary decomposition of \(J(L, M)\) when \(L\) is a distributive lattice and \(M = [2]\). They present some methods to check if a monomial prime ideal belongs to \(\text{ass}(J(L, [2]))\), and give an upper bound in terms of combinatorial properties of \(L\) for the height of the minimal primes. It is also proved that if each minimal prime ideal of \(J(L, [2])\) has height at most three, then \(L\) is a planar lattice and \(\text{width}(L)\leq 2\). Finally, the authors compute the minimal primary decomposition when \(L = [m]\times [n]\) and \(M = [2]\).
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 05E40
Mathematics Subject Classification ID: 13P25
zbMATH DE Number: 6918065
zbMATH Keywords: ideal of lattice homomorphism
zbMATH Keywords: distributive lattice
zbMATH Keywords: monomial ideal
zbMATH Keywords: primary decomposition of ideals
reviewed by: S. A. Seyed Fakhari
MaRDI profile type: Publication
cites work: Q4317713
cites work: The join-meet ideal of a finite lattice
cites work: Monomial ideals and toric rings of Hibi type arising from a finite poset
cites work: Letterplace and co-letterplace ideals of posets
cites work: Lattice Theory: Special Topics and Applications
cites work: Monomial Ideals
cites work: Distributive lattices, bipartite graphs and Alexander duality
cites work: Alexander duality for monomial ideals associated with isotone maps between posets
cites work: Level rings and algebras with straightening laws
cites work: Algebraic properties of ideals of poset homomorphisms
cites work: Q4056048
description: scientific article; zbMATH DE number 6918065