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.
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.
Uniqueness[edit]
Georg Cantor proved that every two densely totally ordered countable sets without lower or upper bounds are order-isomorphic.[1] In particular, there exists an isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proof of this result uses 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[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:
Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation. A dense relation that is also transitive is said to be idempotent.
See also[edit]
References[edit]
- ^ Roitman, Judith (1990), "Theorem 27, p. 123", Introduction to Modern Set Theory, Pure and Applied Mathematics, 8, John Wiley & Sons, ISBN 9780471635192.
- ^ Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN 9781461488545.
Additional reading[edit]
- David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff
