[Paper Review] Convex expansion for finite distributive lattices with applications

Xu Wang, Xuxu Zhao+1 more|arXiv (Cornell University)|Oct 16, 2018

Advanced Algebra and Logic20 references4 citations

TL;DR

This paper introduces the concept of a 'cutting' in finite distributive lattices, enabling a novel convex expansion method that generalizes previous constructions. By leveraging the fundamental theorem of finite distributive lattices, the authors derive a rank generating function, enumerate convex Boolean sublattices and particular elements, and verify the generalized Euler formula for polyhedrons via antichain structures in the meet-irreducible poset.

ABSTRACT

The concept of cutting is first explicitly introduced. By the concept, a convex expansion for finite distributive lattices is considered. Thus, a more general method for drawing the Hasse diagram is given, and the rank generating function of a finite distributive lattice is obtained. In addition, we have several enumerative properties on finite distributive lattices and verify the generalized Euler formula for polyhedrons.

Motivation & Objective

• To formalize and generalize the concept of convex expansion in finite distributive lattices using the novel notion of 'cutting'.
• To provide a unified method for constructing Hasse diagrams of filter lattices and related structures.
• To derive the rank generating function and enumerate convex Boolean sublattices and specific elements in finite distributive lattices.
• To verify the generalized Euler formula for polyhedrons by linking convex Boolean sublattices to antichains in the meet-irreducible poset.

Proposed method

• Introduce 'cutting' as a convex sublattice where every maximal chain intersects it, providing two equivalent characterizations via the fundamental theorem of finite distributive lattices.
• Define a convex expansion operation $ L \boxplus K $ by extending the poset $ P $ associated with $ L = \mathcal{F}(P) $ via a new element $ x_K $ linked to $ S_0 $ and $ S_1 $, the maximal elements below $ \hat{0}_K $ and minimal above $ \hat{1}_K $.
• Establish equivalence between the lattice decomposition $ L = \uparrow\hat{0}_K \cup \downarrow\hat{1}_K $, the cutting condition, and the existence of $ x_K $ such that $ S = P_K * x_K $.
• Use the poset extension to derive recurrence relations for the number of elements of given degree and for the number of convex Boolean sublattices via binomial transforms.
• Apply the framework to Fibonacci and Lucas cubes, deriving recurrence relations for their rank generating functions and enumerative invariants.
• Verify the generalized Euler formula $ \chi = 1 $ by showing $ Q_L(-1) = 1 $, where $ Q_L(x) $ is the rank generating function, and relate it to the number of meet-irreducible elements.

Experimental results

Research questions

• RQ1 How can the convex expansion of finite distributive lattices be generalized beyond prior constructions?
• RQ2 What is the precise algebraic and combinatorial characterization of a 'cutting' in a finite distributive lattice?
• RQ3 How do the rank generating function and the number of convex Boolean sublattices relate to the structure of the meet-irreducible poset?
• RQ4 Can the generalized Euler formula for polyhedrons be verified through lattice-theoretic invariants?

Key findings

• The paper establishes that a sublattice $ K $ is a cutting of $ L $ if and only if $ L = \uparrow\hat{0}_K \cup \downarrow\hat{1}_K $, or equivalently, if $ z < y $ for all $ z \in S_0 $, $ y \in S_1 $, where $ S_0 $ and $ S_1 $ are defined from the poset $ P $ with $ L = \mathcal{F}(P) $.
• The convex expansion $ L \boxplus K $ is constructed by adding a new element $ x_K $ to the poset $ P $, covering all elements in $ S_0 $ and covered by all elements in $ S_1 $, yielding a new lattice with a well-defined Hasse diagram.
• The number of convex Boolean sublattices of $ L $, denoted $ q_k(L) $, satisfies $ q_k(L) = \sum_{j=k}^{m} \binom{j}{k} d_j^{-}(L) $, where $ d_j^{-}(L) $ is the number of maximal antichains of size $ j $ in $ \mathop{\mathrm{Mi}}(L) $.
• The rank generating function $ Q_L(x) $ satisfies $ Q_L(x) = D_L^{-}(1+x) $, where $ D_L^{-}(x) $ is the generating function for the number of maximal antichains of each size in $ \mathop{\mathrm{Mi}}(L) $.
• The generalized Euler formula $ \chi = 1 $ is verified via $ Q_L(-1) = 1 $, which holds because $ D_L^{-}(0) = 1 $, corresponding to the unique minimal element in the lattice.
• For Fibonacci cubes $ \Gamma_n $ and Lucas cubes $ \Lambda_{2n} $, the paper derives recurrence relations: $ F_{m+n} = F_m F_{n+1} + F_{m-1} F_n $, and $ L_{2n} = F_{2n+1} + F_{2n-1} $, respectively.