PreprintCC BY 4.0arXiv
Rank identifications in join-meet ideals of thin distributive lattices
Abstract
Let L be a thin distributive lattice of rank m+1, and let IL be its join-meet ideal in the polynomial ring SL. For A⊆{1,…,m}, let JA be the ideal obtained from IL by identifying, for every i∈ A, the variables of the two elements of L of rank i. We prove that JA is prime if and only if A contains no turning rank and does not contain both ranks adjacent to a turning rank. In this case, SL/JA is the toric ring obtained from Hibi's parametrization of the Hibi ring of L by identifying suitable pairs of variables, and for an arbitrary A we describe the saturation of JA as a lattice ideal.
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BibTeX
@misc{hibi2026rank,
title = {{Rank identifications in join-meet ideals of thin distributive lattices}},
author = {Takayuki Hibi and Francesco Navarra and Ayesha Asloob Qureshi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32647},
eprint = {2610.32647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32647},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rank identifications in join-meet ideals of thin distributive lattices
AU - Takayuki Hibi
AU - Francesco Navarra
AU - Ayesha Asloob Qureshi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32647
UR - https://arxiv.org/abs/2610.32647
AB - Let L be a thin distributive lattice of rank m+1, and let I_L be its join-meet ideal in the polynomial ring S_L. For A⊆{1,…,m}, let J_A be the ideal obtained from I_L by identifying, for every i∈ A, the variables of the two elements of L of rank i. We prove that J_A is prime if and only if A contains no turning rank and does not contain both ranks adjacent to a turning rank. In this case, S_L/J_A is the toric ring obtained from Hibi's parametrization of the Hibi ring of L by identifying suitable pairs of variables, and for an arbitrary A we describe the saturation of J_A as a lattice ideal.
ER -