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 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:

$\forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz\land zRy).$

Every reflexive relation is dense. 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
• Dense-in-itself
• Kripke semantics

References

1. Roitman, Judith (1990), "Theorem 27, p. 123", Introduction to Modern Set Theory, Pure and Applied Mathematics, 8, John Wiley & Sons, ISBN 9780471635192.
2. Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN 9781461488545.

Additional reading

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