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Lattice bases of planar lattices. (English) Zbl 1396.13026

This paper studies planar lattices and the Hibi ideal of a finite distributive lattice. Let \(\mathbb N^2\) denote the distributive lattice consisting of all pairs \((i,j)\) of non-negative integers with the partial order \((i,j) \leq (k,l) \iff i \leq k, j \leq l\). A planar distributive lattice is then a finite sublattice \(\mathcal L\) of \(\mathbb N^2\) containing \((0,0)\) such that for any \((i,j), (k,l) \in \mathcal L\) with \((i,j) < (k,l)\), there exists a chain of \(\mathcal L\) of the form \((i,j) = (i_0, j_0) < (i_1, j_1)< \cdots<(i_s, j_3) = (k,l)\) where \(i_{k+1} + j_{k+1} = i_k + j_k + 1\) for all \(k\).
If \(P\) is a finite partially ordered set (i.e., poset), then we define a poset ideal of \(P\) to be a subset \(Q \subseteq P\) such that if \(\alpha \in Q\) and \(\beta \in P\) such that \(\beta \leq \alpha\) in \(P\) then \(\beta \in Q\). Given such a poset \(P\), we set \(\mathcal I(P)\) to be the finite poset which consists of all poset ideals of \(P\) which are ordered by inclusion. Note that results found in [R. P. Stanley, Enumerative Combinatorics, Vol. I. Monterey, CA: Wadsworth & Brooks/Cole Advanced Books & Software (1986; Zbl 0608.05001)] shows that \(\mathcal I(P)\) is a finite distributive lattice. The author first characterizes when \(\mathcal I(P)\) is planar via the following lemma:
Lemma: With \(P\) and \(\mathcal I(P)\) as above, the following are equivalent:
1.
\(\mathcal I(P)\) is a planar lattice;
2.
\(P\), as a set, is the disjoint union of two chains of \(P\);
3.
If \(p, q, r \in P\), then two of them are compatible.

The second focus of the paper is the Hibi ideal. This part of the discussion involves lattice ideals. Let \(L \subseteq \mathbb Z^n\) be a lattice. Then the lattice ideal attached to \(L\) is the binomial ideal \(I_L \subseteq K[x_1, \ldots, x_n]\) (where \(K\) is a field) generated by \({\mathbf x}^{{\mathbf v}^+} - {\mathbf x^{{\mathbf v}^-}}\) where \({\mathbf v} \in L\), \({\mathbf v}^+\) is the vector obtained from \({\mathbf v}\) by replacing all negative entries by zero and \({\mathbf v}^- = ({\mathbf v} - {\mathbf v}^+)\). Given a finite distributive lattice \(\mathcal L\) and working in the polynomial ring \(K[\{{\mathbf x}_{\alpha}\}_{\alpha \in \mathcal L}]\), we define the Hibi ideal as \(I_{\mathcal L} = ({\mathbf x}_{\alpha}{\mathbf x}_{\beta} - {\mathbf x}_{\alpha \wedge \beta}{\mathbf x}_{\alpha \vee \beta})\). It turns out that \(I_{\mathcal L}\) can be treated as some sublattice of \(\mathbb Z^{|\mathcal L|} = \bigoplus_{\alpha \in \mathcal{L}} \mathbb Ze_{\alpha}\) where \(e_{\alpha}\) is a standard basis for any \(\alpha \in \mathcal L\). For \(\alpha, \beta \in \mathcal L\), the author defines the set \[ \mathcal B = \{e_{\alpha} + e_{\beta} - e_{\alpha \wedge \beta} - e_{\alpha \vee \beta} \mid S_{(\alpha \vee \beta)} \,\,\text{is a square}\} \] where \(S_{(\alpha \vee \beta)}\) is a special sublattice. With this notation, the author shows the following:
Theorem: Let \(\mathcal L\) be a finite distributive lattice whose subposet of joint-irreducible elements is \(P\), and let \(I_{\mathcal L}\) be its Hibi ideal with associated lattice ideal \(I_L\) as discussed above. We have that \(\mathcal B\) is the lattice basis of \(L\) if and only if \(\mathcal L\) is a planar lattice.
The paper concludes by describing the Gröbner basis of the lattice basis ideal attached to the planar lattices via even cycles of an associated graph.

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
18B35 Preorders, orders, domains and lattices (viewed as categories)
03G10 Logical aspects of lattices and related structures

Citations:

Zbl 0608.05001
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