Left Modular Lattices: Theory & Applications

• Left modular lattices are finite bounded lattices with a maximal chain where every element satisfies a modular condition, generalizing distributive behaviors.
• They employ explicit edge-labeling techniques that establish EL-shellability and Cohen–Macaulay properties in their order complexes.
• Their structural framework connects semidistributivity, extremality, and congruence uniformity, with significant applications in group theory and representation.

A left modular lattice is a finite bounded lattice equipped with a maximal chain whose elements satisfy a modularity condition from the left. Concretely, an element $a$ is left modular if for all $b<c$, the equality $(b \vee a) \wedge c = b \vee (a \wedge c)$ holds. Such lattices admit strong combinatorial and topological properties, including EL-shellability and Cohen–Macaulay type order complexes. Left modularity unifies and generalizes key distributivity-like behaviors across lattice theory, with deep connections to extremality, semidistributivity, congruence uniformity, and applications in representation theory and algebraic group theory.

1. Formal Definition and Characterizations

Let $(L, \le)$ be a finite, bounded lattice. A pair $(x, y)$ is a modular pair if for every $z \ge y$,

$(y \vee x) \wedge z = y \vee (x \wedge z).$

An element $x$ is left modular if $(x, y)$ is a modular pair for every $y \in L$ (Woodroofe, 2011), equivalently, for every cover $y \lessdot z$, either $(y \vee x) \wedge z = y$ or $(y \vee x) \wedge z = z$. The modular element (sometimes called two-sided modular) requires the analogous property for $(y, x)$.

A chain $\hat{0} = m_0 < m_1 < \cdots < m_r = \hat{1}$ is left modular if each $m_i$ is left modular; a lattice admitting at least one such maximal chain is called left modular (Mühle, 2021, Segovia, 7 Oct 2025). For join-semidistributive lattices, left modularity has a tight relation with join-extremality (Mühle, 2021); an element $a$ avoids certain pentagon sublattice $N_5$ embeddings (cf. condition 3 in (Segovia, 23 Nov 2025)).

2. Edge-Labelling Criteria and EL-Shellability

The existence of a left modular chain in $L$ admits equivalent edge-labelling characterizations, which immediately yield shellability and topological control (Segovia, 23 Nov 2025). Given a chain $\phi: \hat{0} = x_0 < x_1 < \cdots < x_k = \hat{1}$, let $\delta(j)$ be the smallest $i$ with $j \le x_i$ (for $j$ join-irreducible), and $\beta(m)$ the largest $i$ with $m \ge x_{i-1}$ (for $m$ meet-irreducible).

For any cover $b \lessdot c$, define four labellings: $$\begin{aligned} \gamma_1(b\lessdot c) &= \min\{\delta(j) \mid j \le c,\, j \not\le b\},\ \gamma_1'(b\lessdot c) &= \max\{i \mid c \wedge x_{i-1} \le b\},\ \gamma_2(b\lessdot c) &= \max\{\beta(m) \mid m \ge b,\, m \not\ge c\},\ \gamma_2'(b\lessdot c) &= \min\{i \mid b \vee x_i \ge c\}. \end{aligned}$$ Theorem 3.12 (Segovia, 23 Nov 2025) asserts $\gamma_2 = \gamma_2' \le \gamma_1 = \gamma_1'$ with equality $\gamma_1' = \gamma_2'$ if and only if every $x_i$ on $\phi$ is left modular. When these labelings coincide, they constitute an EL-labelling; thus, the existence of a left modular chain directly yields shellability (Mühle, 2021, Segovia, 7 Oct 2025).

3. Structural Relationships: Semidistributivity, Extremality, Congruence Uniformity

Semidistributivity and extremality occupy a central place in the theory of left modular lattices. The Thomas–Williams theorem establishes that every semidistributive extremal lattice is left modular (Mühle, 2021, Segovia, 23 Nov 2025). Moreover, Mühle proves the converse: every join-semidistributive left modular lattice is join-extremal (Mühle, 2021). For congruence uniform lattices, extremality, left modularity, EL-labelling, and shellability are all equivalent (Segovia, 23 Nov 2025). Distributive lattices are entirely left modular due to the absence of $N_5$ sublattices (Segovia, 23 Nov 2025, Mühle, 2021).

4. Day Doubling, Congruence Normal Lattices, and Uniform Testing

Congruence normal lattices arise via successive Day doublings of convex subsets. The main criterion (Segovia, 23 Nov 2025, Segovia, 7 Oct 2025) for a congruence normal lattice $L = E[C_1, \dots, C_n]$ to be left modular is that at each doubling step $C_i$, the heart $H(C_i)$ must meet some maximal left modular chain of the intermediate lattice $E[C_1,...,C_{i-1}]$,

$$H(C) = \{x \in C \mid x \le \min\{\text{maximal of } C\},\; x \ge \max\{\text{minimal of } C\}\}.$$

In the congruence uniform case, it suffices that each $C_i$ intersects the "spine," i.e., a maximal chain (Segovia, 23 Nov 2025, Segovia, 7 Oct 2025).

The table below organizes these relationships for congruence-normal lattices:

Construction Step	Left Modular Condition	Extremality Condition
Arbitrary Doubling (convex $C$)	Heart $H(C)$ hits left modular chain	n/a
Doubling Intervals (uniform)	Each $C$ hits spine	Each $C$ hits spine

5. Topological Properties: Shellability and Depth

Left modular chains are foundational in demonstrating shellability and computing depth bounds for order complexes. For a graded lattice admitting a left modular chain of length $r$, the $(r-2)$-skeleton of its order complex is vertex-decomposable (and hence shellable): $\mathrm{skel}_{r-2}\,|L|$ is vertex-decomposable (Woodroofe, 2011). If the left modular chain alternates in a maximal chain of length $2r$, the entire lattice is shellable. Shellability implies Cohen–Macaulayness and precise control of Möbius invariants.

The applicability extends to subgroup lattices of finite groups: for a solvable group $G$ with chief length $r$, the subgroup lattice $L(G)$ admits a modular chain yielding $\mathrm{skel}_{r-2}|L(G)|$ vertex-decomposable (and all of $|L(G)|$ shellable) (Woodroofe, 2011). This establishes a topological characterization of solvability: $G$ is solvable iff $\mathrm{depth}|L(G)| \le r-2$.

6. Applications and Examples: $(P,\phi)$-Tamari Lattices, Higher Torsion Class Lattices

$(P,\phi)$-Tamari lattices exemplify the left modular paradigm in a broad family arising from posets $P$ and chains $\phi$ (Segovia, 7 Oct 2025). For such $L = \Tam(P,\phi)$, edge-labelling techniques as in Theorem 3.2 guarantee left modularity, and thus shellability and strong semidistributive properties. Higher torsion class lattices for higher Auslander and Nakayama algebras of type $\mathbb A$ can be realized as $(P,\phi)$-Tamari lattices and inherit left modularity, join-congruence uniformity, and extremality.

For $d,n \ge 1$, the $d$-torsion class lattice $L_n^d$ for the higher Auslander algebra of type $A_n$ is isomorphic to a $(P,\phi)$-Tamari lattice where $P$ is the product poset $os_n^d$ and $\phi(k) = (k,\dots,k)$, with $L_n^d$ possessing left modular, join-semidistributive, join-congruence uniform, and extremal properties.

7. Homotopical Aspects: Discrete Morse Theory, Weakly Descending Chains

The homotopy type of the order complex of a left modular lattice is determined via Babson–Hersh lexicographic discrete Morse theory, applied to the EL-labelling induced by the left modular chain (Woodroofe, 2011). Critical cells arise precisely from weakly descending chains, with strict ascent intervals creating Morse matchings that reduce the complex to a CW-complex with cells indexed by such chains. For modular lattices, this recovers Thévenaz’s result: the order complex is a wedge of $(r-2)$-spheres indexed by chains of complements to a chief series.

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Left modular lattices unify major distributivity-like phenomena in the combinatorial, algebraic, and topological theory of finite lattices, with explicit labelling criteria for structure recognition and generative constructions. The commutativity of shellability, extremality, and congruence uniformity for large classes enables uniform testing, depth control, and broad applicability, particularly in the topology of group lattices and representation-theoretic contexts (Woodroofe, 2011, Mühle, 2021, Segovia, 23 Nov 2025, Segovia, 7 Oct 2025).