Symmetric chain decomposition of necklace posets

TL;DR

This paper proves that the property of being a symmetric chain order is preserved under a specific cyclic quotient operation on product posets, expanding understanding of symmetric chain decompositions.

Contribution

It introduces a new result showing that the quotient of a symmetric chain order by a cyclic group action remains a symmetric chain order.

Findings

Proves $P^n/\mathbb{Z}_n$ is symmetric chain order if $P$ is.

Extends symmetric chain decomposition theory to quotient posets.

Provides tools for analyzing symmetry in poset structures.

Abstract

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $P$ is any symmetric chain order, we prove that $P^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $P^n$ by cyclic permutation of the factors.