Peckness of Edge Posets

TL;DR

This paper introduces a new graded poset based on the edges of a given poset and conjectures its Peck property under group actions, proving it for certain classes of actions and exploring their algebraic properties.

Contribution

It defines the edge poset construction, formulates a conjecture about its Peck property under group actions, and proves it for common cover transitive actions, including their algebraic closure properties.

Findings

Conjecture that $ ext{E}(B_n/G)$ is Peck for group actions on boolean algebras.

Proved the conjecture for common cover transitive actions.

Identified that these actions are closed under direct and semidirect products.

Abstract

For any graded poset $P$, we define a new graded poset, $\mathcal E(P)$, whose elements are the edges in the Hasse diagram of P. For any group, $G$, acting on the boolean algebra, $B_n$, we conjecture that $\mathcal E(B_n/G)$ is Peck. We prove that the conjecture holds for "common cover transitive" actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.