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 as an 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 $\mathbb {Z} [x]$ is a densely ordered set.

Proof

For the element $x\in \mathbb {Z} [x]$, due to the Archimedean property, if $x>0$, there exists a largest integer $n<x$ with $n<x<n+1$, and if $x<0$, $-x>0$, and there exists a largest integer $m=-n-1<-x$ with $-n-1<-x<-n$. As a result, $0<x-n<1$. For any two elements $y,z\in \mathbb {Z} [x]$ with $z<y$, $0<(x-n)(y-z)<y-z$ and $z<(x-n)(y-z)+z<y$. Therefore $\mathbb {Z} [x]$ is dense.

On the other hand, the linear ordering on the integers is not dense.

Uniqueness

Georg Cantor proved that every two non-empty dense totally ordered countable sets without lower or upper bounds are order-isomorphic. 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.

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:

$\forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz\land zRy).$ Alternatively, in terms of composition of R with itself, the dense condition may be expressed as R ⊆ R R.

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

• R is reflexive;
• R is coreflexive;
• R is quasireflexive;
• R is left or right Euclidean; or
• R is symmetric and semi-connex and X has at least 3 elements.

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.

See also

• Dense set — a subset of a topological space whose closure is the whole space
• Dense-in-itself — a subset of a topological space without no isolated points
• Kripke semantics — a dense accessibility relation corresponds to the axiom $\Box \Box A\rightarrow \Box A$

Further reading

• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff