Primary decomposition of ideals of lattice homomorphisms

Leila Sharifan¹ ; Ali Akbar Estaji¹ ; Ghazaleh Malekbala¹

¹Hakim Sabzevari University

The electronic journal of combinatorics, Tome 25 (2018) no. 3

Résumé

For two given finite lattices $L$ and $M$, we introduce the ideal of lattice homomorphism $J(L,M)$, whose minimal monomial generators correspond to lattice homomorphisms $\phi : L\to M$. We 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 $I(L,M)$. Next, we study the minimal primary decomposition of $J(L,M)$ when $L$ is a distributive lattice and $M=[2]$. We present some methods to check if a monomial prime ideal belongs to $\mathrm{ass}(J(L,[2]))$, and we give an upper bound in terms of combinatorial properties of $L$ for the height of the minimal primes. We also show that if each minimal prime ideal of $J(L,[2])$ has height at most three, then $L$ is a planar lattice and $\mathrm{width}(L)\leq 2$. Finally, we compute the minimal primary decomposition when $L=[m]\times [n]$ and $M=[2]$.

Détail

DOI : 10.37236/7694

Classification : 13C05, 05E40, 13P25
Mots-clés : ideal of lattice homomorphism, distributive lattice, monomial ideal, primary decomposition of ideals