Editing Total order (section)

==Chains==

The term '''chain''' is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a [[subset]] of a [[partially ordered set]] that is totally ordered for the induced order.{{sfn|Halmos|1968|loc=Ch.14}}<ref>{{cite book | url=https://www.elsevier.com/books/theory-of-relations/fraisse/978-0-444-50542-2 | isbn=978-0-444-50542-2 | author=Roland Fraïssé | author-link=Roland Fraïssé| title=Theory of Relations | publisher=Elsevier | series=Studies in Logic and the Foundations of Mathematics | volume=145 | edition=1st | date=Dec 2000 }} Here: p. 35</ref> Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term.

A common example of the use of ''chain'' for referring to totally ordered subsets is [[Zorn's lemma]] which asserts that, if every chain in a partially ordered set {{mvar|X}} has an upper bound in {{mvar|X}}, then {{mvar|X}} contains at least one maximal element.<ref>{{cite book | lccn=89009753 | isbn=0-521-36766-2 | author=Brian A. Davey and Hilary Ann Priestley | title=Introduction to Lattices and Order | publisher=Cambridge University Press | series=Cambridge Mathematical Textbooks | year=1990 }} Here: p. 100</ref> Zorn's lemma is commonly used with {{mvar|X}} being a set of subsets; in this case, the upper bound is obtained by proving that the union of the elements of a chain in {{mvar|X}} is in {{mvar|X}}. This is the way that is generally used to prove that a [[vector space]] has [[Hamel bases]] and that a [[ring (mathematics)|ring]] has [[maximal ideal]]s.

In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its [[converse relation|opposite order]]. In this case, a chain can be identified with a [[monotone sequence]], and is called an '''ascending chain''' or a '''descending chain''', depending whether the sequence is increasing or decreasing.<ref>[[Yiannis N. Moschovakis]] (2006) ''Notes on set theory'', [[Undergraduate Texts in Mathematics]] (Birkhäuser) {{ISBN|0-387-28723-X}}, p.&nbsp;116</ref>

A partially ordered set has the [[descending chain condition]] if every descending chain eventually stabilizes.<ref>that is, beyond some index, all further sequence members are equal</ref> For example, an order is [[well-founded order|well founded]] if it has the descending chain condition. Similarly, the [[ascending chain condition]] means that every ascending chain eventually stabilizes. For example, a [[Noetherian ring]] is a ring whose [[ideal (ring theory)|ideals]] satisfy the ascending chain condition.

In other contexts, only chains that are [[finite set]]s are considered. In this case, one talks of a ''finite chain'', often shortened as a ''chain''. In this case, the '''length''' of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain.<ref>Davey and Priestly 1990, Def.2.24, p. 37</ref> Thus a [[singleton set]] is a chain of length zero, and an [[ordered pair]] is a chain of length one. The [[dimension theory|dimension]] of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the [[dimension of a vector space]] is the maximal length of chains of [[linear subspace]]s, and the [[Krull dimension]] of a [[commutative ring]] is the maximal length of chains of [[prime ideal]]s.

"Chain" may also be used for some totally ordered subsets of [[mathematical structure|structures]] that are not partially ordered sets. An example is given by [[regular chain]]s of polynomials. Another example is the use of "chain" as a synonym for a [[walk (graph theory)|walk]] in a [[graph (discrete mathematics)|graph]].