Primary decomposition of ideals of lattice homomorphisms
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 \(M = [2]\).
scientific article; zbMATH DE number 6918065