Scattered order

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 which 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 $\mathbb {Q} \times \mathbb {Z}$ .

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.

[1] Egbert Harzheim (2005). "6.6 Scattered sets". Ordered Sets. Springer. pp. 193–201. ISBN 0-387-24219-8.

[2] 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.

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