nLab
Dedekind completion

Dedekind completions

Idea

The Dedekind completion of a linear order LL is a new linear order L¯\widebar{L} that contains suprema for all inhabited bounded subsets, and such that a supremum in LL is still a supremum in L¯\widebar{L}.

While Dedekind completeness was traditionally described in the context of the real numbers, it can be stated for any linear order, although it really works best for dense? and unbounded (without top or bottom) linear orders. Intuitively, a linear order is Dedekind complete if Dedekind cuts don’t give any ‘new’ elements.

Any paragraph containing the string ‘duiq’ is original research (although lower duiqs at least are known in domain theory).

Definitions

Let SS be a set with a dense? linear order <\lt without top or bottom elements.

Definition

A cut in SS is a pair of subsets L,USL, U \subset S of SS that satisfy the following eight properties:

  1. LL is inhabited;
  2. Dually, UU is inhabited;
  3. If x<yLx \lt y \in L, then xLx \in L;
  4. Dually, if x>yUx \gt y \in U, then xUx \in U;
  5. If xLx \in L, then x<yLx \lt y \in L for some yy;
  6. Dually, xUx \in U, then x>yUx \gt y \in U for some yy;
  7. If x<yx \lt y, then xLx \in L or yUy \in U;
  8. If xLx \in L and yUy \in U, then x<yx \lt y.
Definition

The linearly ordered set SS is Dedekind complete if every cut (L,U)(L,U) is of the form

L={xx<a},U={xx>a} L = \{x \;| \;x \lt a\},\; U = \{x \;|\; x \gt a\}

for some unique aSa \in S.

Definition

If TT is also an unbounded dense linear order, SS is Dedekind complete, and we have a universal arrow u:TSu\colon T \to S in the category of linear orders, then SS (equipped with uu) is the? Dedekind completion of TT.

The set of Dedekind cuts of rational numbers –the set of real numbers– is Dedekind complete. In fact, starting with any unbounded dense linearly ordered set SS, the set of Dedekind cuts is isomorphic to the reals as long as SS is a countably infinite set.

The operation of forming the set of Dedekind cuts is idempotent, so the Dedekind completion can be constructed as the set of Dedekind cuts. More precisely, the Dedekind-complete linear orders form a reflective subcategory of the category of dense unbounded linear orders, so that Dedekind completion is a kind of completion in the abstract categorial sense.

Generalisations

One can generalise Dedekind completion from linear orders to quasiorders.

Definition

A duiq (dense unbounded inhabited quasiorder) is a quasiordered set SS such that, given finite (here always meaning Kuratowski-finite) subsets FF and GG of SS such that x<zx \lt z whenever xFx \in F and zGz \in G, we have some yy in SS such that x<y<zx \lt y \lt z whenever xFx \in F and zGz \in G.

Note that, for SS a linear order, SS is a duiq iff SS is dense, unbounded, and inhabited, hence the term ‘duiq’. (Using linearity, we may assume that FF and GG are subsingletons; then two singleton subsets is denseness, one singleton subset and one empty subset is unboundedness, and two empty subsets is inhabitedness.)

Definition

Given a duiq SS, a cut is a pair (L,U)(L,U) of subsets such that:

  1. LL is inhabited (which is a special case of (5) for FF the empty subset);
  2. Dually, UU is inhabited (a special case of (6));
  3. If x<yLx \lt y \in L, then xLx \in L;
  4. Dually, if x>yUx \gt y \in U, then xUx \in U;
  5. If FF is a finite subset of LL (which we may assume inhabited if we include (1)), then for some xLx \in L, every yFy \in F satisfies y<xy \lt x;
  6. Dually, if FF is a finite (inhabited) subset of UU, then for some xUx \in U, every yFy \in F satisfies y>xy \gt x;
  7. If L<x<UL \lt x \lt U and L<y<UL \lt y \lt U, then x=yx = y;
  8. If xLx \in L and yUy \in U, then x<yx \lt y.

We then define Dedekind-complete duiqs and Dedekind completions of duiqs the same as for dense linear orders, using this notion of cut.

A good example of a duiq is the set of rational-valued functions on any set XX; the Dedekind completion is the set of real-valued functions on XX.

Sections 4.31–39 of HAF do things in even more generality, but I don't really understand it yet.

One-sided Dedekind completions

At least in classical mathematics, considering only LL (for a lower cut) or UU (for an upper cut) doesn't really give us anything new for linear orders; we have only the technicality that (S,)\infty \coloneqq (S,\empty) or (,S)-\infty \coloneqq (\empty,S) is a cut (depending on the side), and we can rule even these out by simply requiring that LL have an upper bound or that UU have a lower bound.

In constructive mathematics, one-sided cuts are more general; see one-sided real number for a discussion of the case where SS is the linear order of rational numbers.

Even in classical mathematics, one-sided cuts do give us something new for quasiorders. Here, we have first more general one-sided notions of duiqs: a lower duiq need only satisfy the condition of a duiq for GG a singleton, and an upper duiq need only satisfy the condition for FF a singleton. Then the lower Dedekind completion of a lower duiq is its set of lower cuts, and the upper Dedekind completion of an upper duiq is its set of upper cuts.

For example, let XX be a compactum and let SS be the quasiordered set of continuous real-valued functions on XX. Then SS is a duiq, hence both a lower and upper duiq. Its lower Dedekind completion is the set of lower semicontinuous functions on XX taking values in the lower reals (which classically are all either real or \infty); and dually on the upper side. Even working classically and ignoring the technicality of \infty, semicontinuous functions are much more general than continuous ones.

References

Revised on October 1, 2012 07:02:12 by Toby Bartels (98.23.147.236)