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 $x<z<y$ . 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.

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Example

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 $\mathbb {Z} [x]$ is a densely ordered set.

Proof

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

On the other hand, the linear ordering on the integers is not dense.

Uniqueness for total dense orders without endpoints

Georg Cantor proved that every two non-empty dense totally ordered countable sets without lower or upper bounds are order-isomorphic.^[1] 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 numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proofs of these results use 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).

Alternatively, in terms of composition of R with itself, the dense condition may be expressed as R ⊆ (R ; R).^[3]

Sufficient conditions for a binary relation R on a set X to be dense are:

R is reflexive;
R is coreflexive;
R is quasireflexive;
R is left or right Euclidean; or
R is symmetric and semi-connex and X has at least 3 elements.

None of them are necessary. For instance, there is a relation R that is not reflexive but dense. A 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 is said to be idempotent.

References

^ Roitman, Judith (1990), "Theorem 27, p. 123", Introduction to Modern Set Theory, Pure and Applied Mathematics, vol. 8, John Wiley & Sons, ISBN 9780471635192.
^ Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN 9781461488545.
^ Gunter Schmidt (2011) Relational Mathematics, page 212, Cambridge University Press ISBN 978-0-521-76268-7

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Ordered vector space Partially ordered Positive cone Riesz space Upper set Young's lattice

This page was last edited on 24 March 2024, at 13:42

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