In mathematical
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, a scattered order is a
linear order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
that contains no
densely ordered subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
with more than one element.
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
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...
.
Laver's theorem Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of ...
(generalizing a conjecture of
Roland Fraïssé
Roland Fraïssé (; 12 March 1920 – 30 March 2008) was a French mathematical logician.
Fraïssé received his doctoral degree from the University of Paris in 1953. In his thesis, Fraïssé used the back-and-forth method to determine whether ...
on
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
orders) states that the embedding relation on the class of countable unions of scattered orders is a
well-quasi-order
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) nor infinite sequenc ...
.
[Harzheim, Theorem 6.17, p. 201; ]
The
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
of a scattered order is
scattered
Scattered may refer to:
Music
* ''Scattered'' (album), a 2010 album by The Handsome Family
* "Scattered" (The Kinks song), 1993
* "Scattered", a song by Ace Young
* "Scattered", a song by Lauren Jauregui
* "Scattered", a song by Green Day from ' ...
. The converse implication does not hold, as witnessed by the
lexicographic order on
.
References
Order theory
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