# 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 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:

- Alternatively, in terms of composition of
*R*with itself, the dense condition may be expressed as*R*⊆*R**R*.[3]

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

## References

- Roitman, Judith (1990), "Theorem 27, p. 123",
*Introduction to Modern Set Theory*, Pure and Applied Mathematics,**8**, John Wiley & Sons, ISBN 9780471635192. - Dasgupta, Abhijit (2013),
*Set Theory: With an Introduction to Real Point Sets*, Springer-Verlag, p. 161, ISBN 9781461488545. - Gunter Schmidt (2011)
*Relational Mathematics*, page 212, Cambridge University Press ISBN 978-0-521-76268-7

## Further reading

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