Dense order

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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.

Example[edit]

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[edit]

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.

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