Dedekind cut

From Academic Kids

In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and xa, then x is in A as well) and B is closed upwards. The cut itself is, conceptually, the "gap" defined between A and B. The original and most important cases are Dedekind cuts for rational numbers and real numbers.

Contents

Handling Dedekind cuts

It is more symmetrical to use the (A,B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one 'half' for example the lower part a. For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval ( −∞, a ), in which case B must be [ a, +∞); or a pair with A the interval ( −∞, a ], in which case B must be ( a, +∞ ).

Ordering Dedekind cuts

If a is a member of S then the set

<math>\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}<math>

is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S.

Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B. In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose.

The cut construction of the real numbers

The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by

<math>A = \{ a\in\textbf{Q} : a^2 < 2 \lor a\le 0 \},<math>
<math>B = \{ b\in\textbf{Q} : b^2 \ge 2 \land b > 0 \}.<math>

This cut represents the real number <math>\sqrt{2}<math> in Dedekind's construction.

Additional structure on the cuts

See construction of real numbers

Generalization: Dedekind completions in posets

More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals.

One completion of S is the set of its downwardly closed subsets (also called order ideals), ordered by inclusion. S is embedded in this lattice by sending each element x to the ideal it generates.

Another completion is obtained by the following construction: For each subset A of S, let Au denote the set of upper bounds of A, and let Al denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind-Macneille completion of S consists of all subsets A for which (Au)l = A; it is ordered by inclusion. The Dedekind-Macneille completion is generally a sublattice of the lattice of order ideals; S is embedded in it in the same way.

Another generalization: surreal numbers

A construction similar to Dedekind cuts is used for the construction of surreal numbers.

See also

fr:coupure de Dedekind

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