Dense order

Density order is a mathematical term from the field of order theory. An order is called dense if between any two elements is a third.

Definition

A linear order < on a set is called tight if

That is, for any two different elements there is a third of which lies between the two.

A subset of a linearly ordered set is called tight if

Specifically, one also says proper proof, to avoid confusion with other leaks terms. The term can be completely analogously define for arbitrary partial orders.

Examples

• The set of rational numbers with the natural order < is dense, as are, it is also a rational number and this is between and.
• The set of real numbers with the natural order < is dense, and those reasons can be performed as for. is properly sealed.
• The set of integers with the natural order < is not tight, there lies between two consecutive integers no third integer.
• By definition, a singleton with the uniquely determined linear order sorted close to it, since there are no two elements for which the defining condition would be met. Some authors conclude from these trivial case, by requiring in addition that the amount must have at least two elements.

Properties

Universal property

By a theorem of Cantor nonempty countable, dense orders without the least and greatest element containing all other countable, linear orders, ie they have the following universal property:

It is a non-empty countable dense linearly ordered set without the least and greatest element and any countable linearly ordered set. Then there is an injective mapping with

Isomorphism classes of countable, dense, linearly ordered sets

After another set of Cantor two non-empty, countable, dense, linearly ordered sets without the least or greatest element is order are. That is: If and two such quantities and both trims are denoted by <, then there exists a bijective mapping with.

The following examples are, therefore, all isomorphic:

• With the natural order
• With the natural order
• With the natural order
• With the natural order
• With the natural order
• With the lexicographic order

When you remove the conditions on the smallest and largest elements, we obtain:

Every countable dense linearly ordered set is isomorphic provided to one of the following six volumes, each with its natural order:

A characterization of the continuum

An order is called complete if every bounded set up has a supremum. After another set of Cantor can the continuum, that is, the set of real numbers, atomic theory, characterized as follows: with the natural order is up to Ordnungsisomorphie the only complete, linear ordering, which is a countable, atomic density and contains too ordnungsisomorphe subset.

Elementary equivalence

The two non-empty dense linear orders without the least and greatest element are elementary equivalent, as can be seen from the set of Fraisse (see here for a proof). In particular, the theories of order and the first-order logic can not distinguish between properties such as completeness can not be in their formulation.