The gold partition conjecture and the Lexicographic sum of posets

TL;DR

This paper investigates how the Gold Partition Conjecture behaves under lexicographic sums of posets, showing that if a base poset satisfies the conjecture, then the sum with any finite poset preserves this property.

Contribution

It proves that the Gold Partition Conjecture is preserved under lexicographic sums with finite posets, extending the conjecture's applicability.

Findings

The Gold Partition Conjecture holds for lexicographic sums if it holds for the base poset.

Lexicographic sums with finite posets preserve the Gold Partition Conjecture.

The result applies to any finite poset added in the lexicographic sum.

Abstract

If a finite poset $Q$ satisfies the Gold Partition Conjecture, and $P$ is a finite poset, then for any $i$ in $P$ the lexicographic sum of $P$ with $Q$ on the point $i$, satisfies the Gold Partition Conjecture.