The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real numbers, the dyadic rationals and the decimal fractions.
In fact, every Archimedean ordered ring extension of the integers
is a densely ordered set.
, due to the Archimedean property, if
, there exists a largest integer
, and there exists a largest integer
On the other hand, the linear ordering on the integers is not dense.
Georg Cantor proved that every two non-empty dense totally ordered countable sets without lower or upper bounds are order-isomorphic.
[1] This makes the theory of dense linear orders without bounds an example of an ω-categorical theory where ω is the smallest limit ordinal.
For example, there exists an order-isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers.
The proofs of these results use the back-and-forth method.
[2] Minkowski's question mark function can be used to determine the order isomorphisms between the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals.
Formally: Sufficient conditions for a binary relation R on a set X to be dense are: None of them are necessary.
For instance, there is a relation R that is not reflexive but dense.
A non-empty and dense relation cannot be antitransitive.
A strict partial order < is a dense order if and only if < is a dense relation.
A dense relation that is also transitive is said to be idempotent.