# Dense order

In mathematics, a partial order or total order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.

## Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. On the other hand, the ordinary (strict) ordering on the integers is not dense.

## Uniqueness

Georg Cantor proved that every two densely totally ordered countable sets without lower or upper bounds are order-isomorphic.[1] In particular, there exists an order-isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proof of this result uses 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.

## Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related. Formally:

${\displaystyle \forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz\land zRy).}$ Alternatively, in terms of composition of R with itself, the dense condition may be expressed as RRR.[3]

Sufficient conditions for a binary relation R in a set X to be dense are:

None of them are necessary. A non-empty and dense relation cannot be antitransitive.

A strict partial order < is a dense order iff < is a dense relation. A dense relation that is also transitive is said to be idempotent.

• Kripke semantics — a dense accessibility relation corresponds to the axiom ${\displaystyle \Box \Box A\rightarrow \Box A}$