scattered order

{{otheruses4|order theory|the Australian post-punk band|Scattered Order}}

In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element.{{cite book|author=Egbert Harzheim

|title=Ordered Sets|url=https://archive.org/details/orderedsets00harz_675

|url-access=limited

|year=2005|publisher=Springer|isbn=0-387-24219-8|contribution=6.6 Scattered sets|pages=[https://archive.org/details/orderedsets00harz_675/page/n199 193]–201}}

A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orders that contains the singleton orders and is closed under well-ordered and reverse well-ordered sums.

Laver's theorem (generalizing a conjecture of Roland Fraïssé on countable orders) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order.Harzheim, Theorem 6.17, p. 201; {{cite journal|first=Richard|last=Laver|authorlink= Richard Laver |title=On Fraïssé's order type conjecture|journal=Annals of Mathematics|volume=93|year=1971|number=1|pages=89–111|jstor=1970754 | doi = 10.2307/1970754}}

The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order on $\backslash mathbb\; Q\backslash times\backslash mathbb\; Z$.