[Paper Review] On the Poset of Multichains

Henri Mühle|arXiv (Cornell University)|Jun 13, 2015

Advanced Algebra and Logic8 references3 citations

TL;DR

This paper introduces the poset of $m$-multichains of a finite poset $\mathcal{P}$, denoted $\mathcal{P}^{[m]}$, as the set of non-decreasing chains of length $m$ under componentwise order. It establishes that structural and topological properties—such as EL-shellability and contractibility of the order complex—are preserved under this construction, and provides a canonical isomorphism between the poset of $m$-multichains of a distributive lattice and the order ideals of $\mathcal{J}(\mathcal{P}) \times \mathcal{C}_m$, linking the construction to Stanley's theory of $\mathcal{P}$-partitions.

ABSTRACT

In this note we introduce the poset of $m$-multichains of a given poset $\mathcal{P}$. Its elements are the multichains of $\mathcal{P}$ consisting of $m$ elements, and its partial order is the componentwise partial order of $\mathcal{P}$. We show that this construction preserves a number of poset-theoretic and poset-topological properties of $\mathcal{P}$. Moreover, we describe the structure of the poset of $m$-multichains of a finite distributive lattice, and provide a link to R.~Stanley's theory of $\mathcal{P}$-partitions.

Motivation & Objective

• To define and study the poset of $m$-multichains of a finite poset $\mathcal{P}$, denoted $\mathcal{P}^{[m]}$, as a subposet of the $m$-fold product $\mathcal{P}^m$ under componentwise order.
• To investigate which poset-theoretic and topological properties of $\mathcal{P}$ are preserved in $\mathcal{P}^{[m]}$, particularly EL-shellability and contractibility of the order complex.
• To provide a structural characterization of $\mathcal{P}^{[m]}$ when $\mathcal{P}$ is a finite distributive lattice, linking it to order ideals in a product poset.
• To establish a connection between $m$-multichains in distributive lattices and R. Stanley’s theory of $\mathcal{P}$-partitions via a canonical isomorphism.

Proposed method

• Define $\mathcal{P}^{[m]}$ as the set of all $m$-tuples $(x_1, \dots, x_m)$ with $x_1 \leq \cdots \leq x_m$, ordered componentwise.
• Show that $\mathcal{P}^{[m]}$ is an induced subposet of $\mathcal{P}^m$, preserving order relations from the product.
• Prove that if $\mathcal{P}$ is EL-shellable, then $\mathcal{P}^{[m]}$ is also EL-shellable, using a natural edge-labeling construction.
• Demonstrate that the order complex of $\mathcal{P}^{[m]}$ is contractible for $m > 1$, using a deformation retraction argument.
• Apply Birkhoff’s representation theorem to show that for a finite distributive lattice $\mathcal{P}$, the poset $\mathcal{P}^{[m]}$ is isomorphic to the order ideals of $\mathcal{J}(\mathcal{P}) \times \mathcal{C}_m$, where $\mathcal{J}(\mathcal{P})$ is the poset of join-irreducible elements.
• Establish a bijection between $m$-multichains in $\mathcal{P}$ and $\mathcal{P}$-partitions of height at most $m$, linking the construction to Stanley’s theory of $\mathcal{P}$-partitions via the isomorphism $\mathcal{A}(\mathcal{P};m) \cong \mathcal{I}(\mathcal{P} \times \mathcal{C}_m)$.

Experimental results

Research questions

• RQ1 Does the construction of $m$-multichains preserve EL-shellability in the original poset $\mathcal{P}$?
• RQ2 Is the order complex of $\mathcal{P}^{[m]}$ contractible for $m > 1$?
• RQ3 How can the poset of $m$-multichains of a finite distributive lattice be characterized in terms of order ideals?
• RQ4 What is the relationship between $m$-multichains in a distributive lattice and Stanley’s $\mathcal{P}$-partitions?
• RQ5 Can the poset of $m$-multichains be naturally identified with a known combinatorial structure, such as the order ideals of a product poset?

Key findings

• The poset $\mathcal{P}^{[m]}$ of $m$-multichains inherits EL-shellability from $\mathcal{P}$, meaning that if $\mathcal{P}$ admits a consistent edge-labeling, so does $\mathcal{P}^{[m]}$.
• The order complex of $\mathcal{P}^{[m]}$ is contractible for all $m > 1$, implying that $\mathcal{P}^{[m]}$ has the homotopy type of a point.
• For a finite distributive lattice $\mathcal{P}$, the poset $\mathcal{P}^{[m]}$ is isomorphic to the order ideals of $\mathcal{J}(\mathcal{P}) \times \mathcal{C}_m$, where $\mathcal{J}(\mathcal{P})$ is the poset of join-irreducible elements of $\mathcal{P}$.
• The poset $\mathcal{A}(\mathcal{P};m)$ of $m$-bounded $\mathcal{P}$-partitions is isomorphic to $\mathcal{I}(\mathcal{P} \times \mathcal{C}_m)$, and this isomorphism extends to $\mathcal{P}^{[m]} \cong \mathcal{I}(\mathcal{J}(\mathcal{P}) \times \mathcal{C}_m)$ for distributive lattices.
• The number of $m$-multichains in a distributive lattice is strictly less than the number of $m$-bounded $\mathcal{P}$-partitions, as illustrated by examples with 12 vs. 46 and 13 vs. 33 such objects in specific lattices.
• The construction provides a canonical isomorphism between $m$-multichains in a distributive lattice and order ideals in a product of the join-irreducible poset and a chain of size $m$, unifying two combinatorial frameworks.