Representable posets and their order components

TL;DR

This paper investigates the conditions under which a partially ordered set can be represented by prime ideals of a bounded distributive lattice, establishing that representability of components implies the whole, but not vice versa.

Contribution

It proves that if all order components of a poset are representable, then the entire poset is also representable, and provides a counterexample for the converse.

Findings

Representability of components implies the entire poset is representable.

Counterexample shows the converse does not hold.

Provides insight into the structure of representable posets.

Abstract

A partially ordered set P is representable if there is a bounded distributive lattice such that its ordered set of prime ideals is order-isomorphic to P. We show that if the order components of a poset P are representable, then so is P. Moreover, we provide an example disproving the converse.