Algebras with straightening laws on join- or meet-semidistributive lattices

Abstract: We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen--Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen--Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen--Macaulayness are equivalent.

• The paper proves that for join-semidistributive lattices, integrality implies meet-distributivity, while meet-distributivity is equivalent to Cohen–Macaulayness over every field.
• It characterizes meet-distributivity through connectivity of the maximal-chain exchange graph using canonical join-irreducible labels, providing a purely lattice-theoretic method.
• Explicit examples show the implications are strict: the 13-element lattice L₁₃ is meet-distributive but not integral, while the 7-element lattice L₇ is integral but non-distributive.

Background and motivation

Algebras with straightening laws (ASLs) encode the multiplication rules of a graded commutative algebra through the combinatorics of an underlying poset $P$: the standard monomials indexed by chains of $P$ form a $k$-basis, and products of incomparable elements admit straightening relations dominated by lower elements of the poset. A poset is integral if it supports a homogeneous ASL that is a domain. Hibi showed that every distributive lattice is integral and conjectured that every integral lattice is Cohen–Macaulay (2608.17438); the analogous statement for general posets fails by a construction of Terai. The paper under discussion, by Matsushita, Miyashita, and Tani, settles this conjecture affirmatively for join- and meet-semidistributive lattices, and does so via a purely lattice-theoretic characterization of meet- and join-distributivity in terms of the connectivity of the maximal-chain exchange graph.

The main theorem

The central result is a four-part theorem. For a join-semidistributive (JSD) lattice $L$ (dually, meet-semidistributive), the paper establishes:

• Meet-distributivity equals Cohen–Macaulayness: $L$ is meet-distributive (MD) if and only if $L$ is Cohen–Macaulay over an arbitrary field — a strong field-independent statement.
• Integrality implies meet-distributivity: any integral JSD lattice is MD.
• The inclusions are strict in both directions: there exists a JSD lattice that is MD but not integral (the 13-element lattice $L_{13}$), and a JSD lattice that is integral but not distributive (the 7-element lattice $L_7$).

Two corollaries follow immediately. First, Hibi's conjecture holds for JSD and MSD lattices, with the strengthening that Cohen–Macaulayness holds over every field. Second, for fully semidistributive lattices, the three conditions — distributivity, integrality, and Cohen–Macaulayness over an arbitrary field — are equivalent, since a lattice that is both MD and join-distributive is semidistributive and modular, hence distributive.

Connectivity of the maximal-chain exchange graph

The key lattice-theoretic contribution is a characterization of MD-lattices among JSD lattices via the maximal-chain exchange graph $G(L)$, whose vertices are maximal chains and whose edges connect chains differing by exactly one element. The proof rests on the canonical join-irreducible labeling $\lambda$ of a JSD lattice, where $P$0 is the unique minimal $P$1 with $P$2 for a cover $P$3.

Two lemmas drive the argument. A "diamond lemma" shows that the labels on the two parallel edges of any diamond $P$4 satisfy $P$5 and $P$6; this follows from the uniqueness of the canonical join representation. Consequently, the join-label set $P$7 is invariant under adjacency in $P$8, and labels along any chain are pairwise distinct, so $P$9 is constant on connected components of $k$0 and has cardinality equal to the chain length.

The characterization then reads: a JSD lattice is MD if and only if $k$1 is connected (2608.17438). If $k$2 is connected, every maximal chain shares the common label set $k$3, so all maximal chains have length $k$4, which by Czédli's characterization is equivalent to meet-distributivity. Conversely, an MD lattice is JSD with upper semimodular dual, so its order complex is shellable by Björner's theorem, and shellable complexes are strongly connected. For fully semidistributive lattices this yields: $k$5 is connected if and only if $k$6 is distributive.

Proof of the algebraic equivalences

The passage from graph connectivity to Cohen–Macaulayness and integrality uses standard facts about ASLs and their Gröbner degenerations. For any homogeneous ASL $k$7 on $k$8, the discrete ASL $k$9 is a Gröbner degeneration of $L$0, so $L$1 and $L$2 share their $L$3-vector, and $L$4 is Cohen–Macaulay exactly when $L$5 is Cohen–Macaulay. Moreover, if $L$6 is Cohen–Macaulay over some field or integral, then $L$7 is pure and strongly connected, i.e., $L$8 is connected.

Combining these: if $L$9 is MD, shellability of $L$0 gives Cohen–Macaulayness over every field; if $L$1 is Cohen–Macaulay over some field or integral, connectivity of $L$2 forces MD. This proves parts (1) and (2) of the main theorem and the corollary on Hibi's conjecture.

Sharpness: the two examples

The strictness of the implications is established by explicit small lattices.

The 13-element lattice $L$3 is MD (each interval $L$4 is Boolean) but not integral. The obstruction is numerical: Macaulay2 computes $L$5, and since the $L$6-vector is an ASL invariant, any hypothetical ASL domain on $L$7 would carry the same $L$8-vector. But Stanley's theorem on graded Cohen–Macaulay domains requires $L$9, and here $L$0, a contradiction.

The 7-element lattice $L$1 is MD but not distributive, yet it is integral: the map $L$2 sending $L$3 to $L$4 has kernel generated by six binomials that satisfy the straightening axiom and form a Gröbner basis with initial ideal $L$5. Hence $L$6 is a homogeneous ASL domain on $L$7, showing that integrality does not force distributivity even within the semidistributive class.

Limitations and open questions

The results are confined to the semidistributive hierarchy; the paper does not address Hibi's conjecture for general modular or non-semidistributive lattices, where related work by Hibi and Seyed Fakhari on modular lattices is noted as unpublished. The non-integrality of $L$8 is established via an $L$9-vector obstruction specific to Cohen–Macaulay domains, so the argument does not extend directly to ruling out integrality for MD lattices whose $L_{13}$0-vectors happen to satisfy Stanley's inequalities. Whether the equivalence between distributivity and integrality for semidistributive lattices admits a structural (rather than case-by-case) explanation, and whether the exchange-graph connectivity criterion generalizes to broader lattice classes, remain open.

Conclusion

The paper proves that for join-semidistributive lattices, meet-distributivity, Cohen–Macaulayness over every field, and (as a consequence of integrality) are tightly linked: integrality implies meet-distributivity, meet-distributivity implies Cohen–Macaulayness, and neither converse holds. For semidistributive lattices, distributivity, integrality, and Cohen–Macaulayness coincide, confirming Hibi's conjecture in this setting. The technical core — the equivalence of meet-distributivity with connectivity of the maximal-chain exchange graph via the canonical join-labeling — is of independent interest in combinatorial lattice theory.