Extremality in semidistributive lattices

TL;DR

The paper investigates deep interactions between extremality, left modularity, and semidistributivity in finite lattices, establishing edge-labelling criteria that characterize left modularity and linking doubling constructions to extremality. It proves that for congruence uniform lattices extremality, left modularity, and shellability are equivalent, and it provides a dimension formula for semidistributive extremal lattices in terms of the Galois graph’s complement. A central contribution is the dimension-determination tool, which yields exact dimensions for notable lattice families (Hochschild, bubble, (m,n)-word, parabolic Tamari) and illuminates dimensions of torsion-class lattices, thereby connecting lattice theory with representation theory and algebraic combinatorics. The work also furnishes a counterexample to Barnard’s question on induced subcomplexes, and closes with a set of open questions guiding future exploration in this rich intersection of combinatorics and representation theory.

Abstract

We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with necessary and sufficient conditions on the doubling steps in the construction of congruence normal lattices that ensure left modularity or extremality. We prove that a congruence uniform lattice is shellable if and only if it is extremal. We answer a question of Barnard by constructing a counterexample showing that an induced subcomplex of a canonical join complex need not itself be such a complex. Finally, we show that the order dimension of a semidistributive extremal lattice equals the chromatic number of the complement of its Galois graph, generalizing a theorem of Dilworth for distributive lattices. As an application, we determine the dimensions of generalizations of the Hochschild lattice, of the parabolic Tamari lattice, and of some lattices of torsion classes.

Figures (14)

• Figure 1:
• Figure 2: We represent $3$ successive doublings. The left modular elements are the blue dots. The thick red edges form the convex subsets $C$ that we double and we circled the elements of $H(C)$.
• Figure 3:
• Figure 4:
• Figure 5: The Galois graph of a semidistributive lattice having $167$ elements. The arrows are of the three types $\rightarrow$, $\hookrightarrow$ and $\twoheadrightarrow$ as in reading2021fundamental and they form a two-acyclic factorization system.

Theorems & Definitions (140)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3: Thomas_2019
• Proposition 2.4: MarkowskyExtremal