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Join-meet binomial algebras of distributive lattices

Barbara Betti, Takayuki Hibi

Abstract

We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated by quadrics is studied.

Join-meet binomial algebras of distributive lattices

Abstract

We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated by quadrics is studied.
Paper Structure (3 sections, 12 theorems, 17 equations, 7 figures)

This paper contains 3 sections, 12 theorems, 17 equations, 7 figures.

Table of Contents

  1. Partially ordered sets, distributive lattices and Gröbner bases
  2. Algebras with straightening laws
  3. Relations of join-meet binomials

Key Result

Lemma 1.1

Let $f_1, \ldots, f_s$ be homogeneous polynomials of $S=K[x_1, \ldots, x_n]$ of the same degree $>0$ and let $<$ be a monomial order on $S$. Consider the subrings $K[f_1,\ldots,f_s]$ and $K[{\rm in}_<(f_1), \ldots, {\rm in}_<(f_s)]$ of $S$. Let $A=K[y_1, \ldots, y_s]$ be the polynomial ring in $s$ v

Figures (7)

  • Figure 1: Boolean lattice $B_3$ of rank $3$.
  • Figure 2: A thin distributive lattice.
  • Figure 3: The sublattice $Q_L$ of the thin distributive lattice $L$ of Figure \ref{['fig: thin distributive lattice']}
  • Figure 4: Divisor lattice $D_{2^2\cdot 3^2}$.
  • Figure 5: Nonplanar distributive lattice with $\theta(L)=3$
  • ...and 2 more figures

Theorems & Definitions (29)

  • Lemma 1.1
  • proof
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 19 more