Scattered order
Appearance
In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element.[1]
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.[2]
The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order on .
References
- ^ Egbert Harzheim (2005). "6.6 Scattered sets". Ordered Sets. Springer. pp. 193–201. ISBN 0-387-24219-8.
- ^ Harzheim, Theorem 6.17, p. 201; Laver, Richard (1971). "On Fraïssé's order type conjecture". Annals of Mathematics. 93 (1): 89–111. doi:10.2307/1970754. JSTOR 1970754.