Structural theory of trees. II. Completeness and completions of trees

TL;DR

This paper develops a structural theory for trees, exploring various notions of completeness and introducing methods to construct minimal complete trees extending given trees.

Contribution

It extends Dedekind completeness concepts to trees and proposes constructions for minimal completions of trees with these properties.

Findings

Defined several natural notions of completeness for trees

Extended Dedekind completeness to tree structures

Provided methods for constructing minimal complete trees

Abstract

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of \emph{tree completions} that extend any tree to a minimal one satisfying the respective completeness property.