Scattered order explained

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

Q x Z

.

Notes and References

  1. Book: Egbert Harzheim. Ordered Sets. limited. 2005. Springer. 0-387-24219-8. 6.6 Scattered sets. 193–201.
  2. Harzheim, Theorem 6.17, p. 201; Richard. Laver. Richard Laver . On Fraïssé's order type conjecture. Annals of Mathematics. 93. 1971. 1. 89–111. 1970754 . 10.2307/1970754.