definition of well ordered set, a variant

A well-ordered set is normally defined as a totally ordered set in which every nonempty subset has a least member, as the parent object does.

It is possible to define well-ordered sets as follows:

a well-ordered set $X$ is a partially ordered set in which every nonempty subset of $X$ has a least member.

To justify the alternative, we prove that every partially ordered set $X$ in which every nonempty subset has a least member is total:

let $x\in X$ and $y\in X$ , $x\neq y$ . Now, $\{x,y\}$ has a least member, a fortiori, $x, y$ are comparable. Hence, $X$ is totally ordered.

The alternative has the benefit of being a stronger statement in the sense that

(partial\;order)\Longrightarrow(total\;order)

given that every nonempty subset has a least member.

References

1 Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.
2 Jech, T., Set Theory, 3rd millennium ed., Springer, 2002.

Title	definition of well ordered set, a variant
Canonical name	DefinitionOfWellOrderedSetAVariant
Date of creation	2013-03-22 18:04:47
Last modified on	2013-03-22 18:04:47
Owner	yesitis (13730)
Last modified by	yesitis (13730)
Numerical id	9
Author	yesitis (13730)
Entry type	Derivation
Classification	msc 03E25
Classification	msc 06A05
Related topic	SomethingRelatedToNaturalNumber
Related topic	NaturalNumbersAreWellOrdered