Total of each of the 1-th and 2-th Betti numbers of the join-meet ideal of Special Crystal Lattice

TL;DR

This paper computes the Betti numbers of the join-meet ideal of a special Crystal lattice, revealing combinatorial patterns and suggesting a link to monomial orders.

Contribution

It provides the first computational analysis of Betti numbers for join-meet ideals of non-distributive lattices, specifically Crystal lattices.

Findings

Betti numbers are characterized by combinations of generators.

Results suggest a connection between monomial orders and ideal properties.

Computational methods reveal structural insights into the join-meet ideal.

Abstract

The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomial ideals that are defined by finite lattices. We study the join-meet ideal of non-distributive finite lattices that do not always satisfy modular. In particular, we work on the case of Crystal lattice which is one of them. In this paper, we give the computational result of a total of each of the 1-th and 2-th of grade Betti numbers of the join-meet ideal of a special Crystal lattice. The important point about this result is that it is characterized by the number of combinations of special generators. Moreover, we can consider from this result that a compatible monomial order suggests a connection to something good.