Pruning a poset with veins

TL;DR

This paper introduces a novel method called pruning of a poset using veins, which simplifies the structure while preserving key irreducible elements, based on abstract connectivity concepts.

Contribution

It defines veins as special chains in posets and introduces the pruning order, a new partial order that simplifies the poset without losing critical elements.

Findings

Pruned posets are simpler yet retain irreducible elements.

Veins are special chains that meet specific maximal chain conditions.

Pruning preserves key structural elements of the original poset.

Abstract

We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define a new partial order on the same underlying set, called the pruning order. The associated pruned poset is simpler than the initial poset, but irreducible, coirreducible, and doubly-irreducible elements are preserved by the operation of pruning.