Dense order

In mathematics, a partial 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.

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.

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.^{[citation needed]} 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.

See also

• Dense set
• Dense-in-itself
• Kripke semantics

References

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