{{Short description|Type of ordering of a set}} In [[mathematics]], a [[partial order]] or [[total order]] < on a [[Set (mathematics)|set]] <math>X</math> is said to be '''dense''' if, for all <math>x</math> and <math>y</math> in <math>X</math> for which <math>x < y</math>, there is a <math>z</math> in <math>X</math> such that <math>x < z < y</math>. 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 [[comparability|comparable]].

Equivalently, a partial order is dense precisely if its [[covering relation]] is empty.

==Example== The [[rational number]]s as a linearly ordered set are a densely ordered set in this sense, as are the [[algebraic number]]s, the [[real number]]s, the [[dyadic rational]]s and the [[decimal fraction]]s. In fact, every [[Archimedean property|Archimedean]] ordered [[Subring#Ring extensions|ring extension]] of the [[integers]] <math>\mathbb{Z}[x]</math> is a densely ordered set.

{{Math proof|title=Proof|drop=hidden|proof= For the element <math>x\in\mathbb{Z}[x]</math>, due to the Archimedean property, if <math>x > 0</math>, there exists a largest integer <math>n < x</math> with <math>n < x < n + 1</math>, and if <math>x < 0</math>, <math>-x > 0</math>, and there exists a largest integer <math>m = -n - 1 < -x</math> with <math>-n - 1 < -x < -n</math>. As a result, <math>0 < x - n < 1</math>. For any two elements <math>y, z\in\mathbb{Z}[x]</math> with <math>z < y</math>, <math>0 < (x - n)(y - z) < y - z</math> and <math>z < (x - n)(y - z) + z < y</math>. Therefore <math>\mathbb{Z}[x]</math> is dense. }}

On the other hand, the linear ordering on the [[integer]]s is not dense.

==Uniqueness for total dense orders without endpoints == {{main|Cantor's isomorphism theorem}} [[Georg Cantor]] proved that every two non-empty dense totally ordered [[countable set]]s without lower or upper bounds are [[order isomorphism|order-isomorphic]].<ref>{{citation|title=Introduction to Modern Set Theory|volume=8|series=Pure and Applied Mathematics|first=Judith|last=Roitman|authorlink=Judith Roitman|publisher=John Wiley & Sons|year=1990|isbn=9780471635192|url=https://books.google.com/books?id=0euMlBhBxjMC&pg=PA123|contribution=Theorem 27, p.&nbsp;123}}.</ref> This makes the theory of dense linear orders without bounds an example of an ω-[[categorical theory]] where ω is the smallest [[limit ordinal]]. For example, there exists an order-isomorphism between the [[rational number]]s and other densely ordered countable sets including the [[dyadic rational]]s and the [[algebraic number]]s. The proofs of these results use the [[back-and-forth method]].<ref>{{citation|title=Set Theory: With an Introduction to Real Point Sets|first=Abhijit|last=Dasgupta|publisher=Springer-Verlag|year=2013|isbn=9781461488545|page=161|url=https://books.google.com/books?id=u06-BAAAQBAJ&pg=PA161}}.</ref>

[[Minkowski's question mark function]] can be used to determine the order isomorphisms between the [[Quadratic irrational number|quadratic algebraic numbers]] and the [[rational number]]s, and between the rationals and the [[dyadic rational]]s.

==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: : <math> \forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz \land zRy).</math> Alternatively, in terms of [[composition of relations|composition]] of ''R'' with itself, the dense condition may be expressed as ''R'' ⊆ (''R'' ; ''R'').<ref>[[Gunter Schmidt]] (2011) ''Relational Mathematics'', page 212, [[Cambridge University Press]] {{ISBN|978-0-521-76268-7}}</ref>

[[Sufficient condition]]s for a binary relation ''R'' on a set ''X'' to be dense are: * ''R'' is [[reflexive relation|reflexive]]; * ''R'' is [[coreflexive relation|coreflexive]]; * ''R'' is [[quasireflexive relation|quasireflexive]]; * ''R'' is left or right [[Euclidean relation|Euclidean]]; or * ''R'' is [[symmetric relation|symmetric]] and [[semi-connex relation|semi-connex]] and ''X'' has at least 3 elements. None of them are [[necessary condition|necessary]]. For instance, there is a relation R that is not reflexive but dense. A [[Empty relation|non-empty]] 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 relation|transitive]] is said to be [[idempotent relation|idempotent]].

==See also== *[[Dense set]] &mdash; a subset of a topological space whose closure is the whole space *[[Dense-in-itself]] &mdash; a subset <math>A</math> of a topological space such that <math>A</math> does not contain an isolated point *[[Kripke semantics]] &mdash; a dense accessibility relation corresponds to the axiom <math>\Box\Box A \rightarrow \Box A</math>

==References== {{reflist}}

==Further reading==

* [[David Harel]], [[Dexter Kozen]], Jerzy Tiuryn, ''Dynamic logic'', MIT Press, 2000, {{ISBN|0-262-08289-6}}, p.&nbsp;6ff

{{Order theory}}

[[Category:Properties of binary relations]] [[Category:Order theory]]