In mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable.
The rational numbers as a 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 is a densely ordered set.
For the element , due to the Archimedean property, if , there exists a largest integer with , and if , , and there exists a largest integer with . As a result, . For any two elements with , and . Therefore is dense.
On the other hand, the linear ordering on the integers is not dense.
Georg Cantor proved that every two nonempty dense totally ordered countable sets without lower or upper bounds are orderisomorphic.^{[1]} This makes the theory of dense linear orders without bounds an example of an ωcategorical theory where ω is the smallest limit ordinal. For example, there exists an orderisomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proofs of these results use the backandforth 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.
Any binary relation R is said to be dense if, for all Rrelated x and y, there is a z such that x and z and also z and y are Rrelated. Formally:
Sufficient conditions for a binary relation R on a set X to be dense are:
None of them are necessary. For instance, there is a relation R that is not reflexive but dense. A nonempty and dense relation cannot be antitransitive.
A strict partial order < is a dense order if and only if < is a dense relation. A dense relation that is also transitive is said to be idempotent.
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