On the Well Extension of Partial Well Orderings

TL;DR

This paper investigates conditions under which partial well orderings can be extended to well orderings, proving that any such structure can be extended and characterizing when linear extensions are well-ordered.

Contribution

It establishes that any partial well order can be extended to a well order and characterizes when linear extensions are well-ordered based on the absence of infinite totally unordered subsets.

Findings

Any partial well order can be extended to a well order.

Linear extensions are well-ordered iff no infinite totally unordered subset exists.

Provides a necessary and sufficient condition for linear extensions to be well-ordered.

Abstract

In this paper, we study the well extension of strict(irreflective) partial well orderings. We first prove that any partially well-ordered structure <A, R> can be extended to a well-ordered one. Then we prove that every linear extension of <A, R> is well-ordered if and only if A has no infinite totally unordered subset under R.