Scattered order
From HandWiki
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 [math]\displaystyle{ \mathbb Q\times\mathbb Z }[/math].
References
- ↑ Egbert Harzheim (2005). "6.6 Scattered sets". Ordered Sets. Springer. pp. 193–201. ISBN 0-387-24219-8. https://archive.org/details/orderedsets00harz_675.
- ↑ 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.
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