nLab
well-founded relation

Contents

Idea

A (binary) relation \prec on a set SS is called well-founded if it is valid to do induction on \prec over SS.

Definition

Given a subset AA of SS, suppose that AA has the property that, given any element xx of SS, if

(t:S),txtA, \forall (t\colon S),\; t \prec x \Rightarrow t \in A ,

then xAx \in A. Such an AA may be called a \prec-inductive subset of SS. The relation \prec is well-founded if the only \prec-inductive subset of SS is SS itself.

Note that this is precisely what is necessary to validate induction over \prec: if we can show that a statement is true of xSx\in S whenever it is true of everything \prec-below xx, then it must be true of everything in SS. In the presence of excluded middle it is equivalent to other commonly stated definitions; see Formulations in classical logic below.

Formulations in classical logic

While the definition above follows how a well-founded relation is generally used (namely, to prove properties of elements of SS by induction), it is complicated. Two alternative formulations are given by the following:

  1. The relation \prec has no infinite descent (usually attributed to Pierre de Fermat) if there exists no sequence x 2x 1x 0\cdots \prec x_2 \prec x_1 \prec x_0 in SS. (Such a sequence is called an infinite descending sequence.)

  2. The relation \prec is classically well-founded if every inhabited subset AA of SS has a member xAx \in A such that no tAt \in A satisfies txt \prec x. (Such an xx is called a minimal element of AA.)

In classical mathematics, both of these conditions are equivalent to being well-founded. Constructively, we may prove that a well-founded relation has no infinite descent, but not the converse; and we may prove that a classically well-founded relation is well-founded, but not the converse. (In fact, if there exists an inhabited relation that is classically well-founded, then excluded middle follows.) In predicative mathematics, however, the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable (if classical logic is used).

Even in constructive predicative mathematics, (1) is strong enough to establish the Burali-Forti paradox (when applied to linear orders). In material set theory, (2) is traditionally used to state the axiom of foundation, although the impredicative definition could also be used as an axiom scheme (and must be in constructive versions). In any case, either (1) or (2) is usually preferred by classical mathematicians as simpler.

Coalgebraic formulation

Many inductive or recursive notions may also be packaged in coalgebraic terms. For the concept of well-founded relation, first observe that a binary relation \prec on a set XX is the same as a coalgebra structure θ:XP(X)\theta\colon X \to P(X) for the covariant power-set endofunctor on SetSet, where yxy \prec x if and only if yθ(x)y \in \theta(x).

In this language, a subset i:UXi\colon U \hookrightarrow X is \prec-inductive, or θ\theta-inductive, if in the pullback

H j X θ PU Pi PX\array{ H & \stackrel{j}{\to} & X \\ \downarrow & & \downarrow^\mathrlap{\theta} \\ P U & \underset{P i}{\to} & P X }

the map jj factors through ii. (Note that jj is necessarily monic, since PP preserves monos.) Unpacking this a bit: for any xXx \in X, if θ(x)=V\theta(x) = V belongs to PUP U, that is if θ(x)U\theta(x) \subseteq U, then xUx \in U. This says the same thing as x:X( y:XyxyU)xU\forall_{x\colon X} (\forall_{y\colon X} y \prec x \Rightarrow y \in U) \Rightarrow x \in U.

Then, as usual, the PP-coalgebra (X,θ)(X, \theta) is well-founded if every θ\theta-inductive subset UXU \hookrightarrow X is all of XX.

Other relevant notions may also be packaged; for example, the PP-coalgebra XX is extensional if θ:XPX\theta\colon X \to P X is monic. See also well-founded coalgebra.

Simulations

Given two sets SS and TT, each equipped with a well-founded relation \prec, a function f:STf\colon S \to T is a simulation of SS in TT if

  1. f(x)f(y)f(x) \prec f(y) whenever xyx \prec y and
  2. given tf(x)t \prec f(x), there exists yxy \prec x with t=f(y)t = f(y).

Then sets so equipped form a category with simulations as morphisms. See extensional relation for more uses of simulations.

In coalgebraic language, a simulation STS \to T is simply a PP-coalgebra homomorphism f:STf\colon S \to T. Condition (1), that ff is merely \prec-preserving, translates to the condition that ff is a colax morphism of coalgebras, in the sense that there is an inclusion

X θ X PX f Pf Y θ Y PY.\array{ X & \stackrel{\theta_X}{\to} & P X \\ ^\mathllap{f} \downarrow & \swArrow & \downarrow^\mathrlap{P f} \\ Y & \underset{\theta_Y}{\to} & P Y. }

Properties

Every well-founded relation is irreflexive; that is, xxx \nprec x. Sometimes one wants a reflexive version \preceq of a well-founded relation; let xyx \preceq y if and only xyx \prec y or x=yx = y. Then the requirement that xx be a minimal element of a subset AA states that txt \preceq x only if t=xt = x. But infinite descent or direct proof by induction still require \prec rather than \preceq.

A well order may be defined as a well-founded linear order, or alternatively as a transitive, extensional, well-founded relation.

A well-quasi-order is a well-founded preorder (referring to the reflexive version of well-foundedness above) that in addition has no infinite antichains.

The axiom of foundation in material set theory states precisely that the membership relation \in on the proper class of all pure sets is well-founded. In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.

Examples

Let SS be a finite set. Then any relation on SS whose transitive closure is irreflexive is well-founded.

Let SS be the set of natural numbers, and let xyx \prec y if yy is the successor of xx: y=x+1y = x + 1. That this relation is well-founded is the usual principle of mathematical induction.

Again let SS be the set of natural numbers, but now let xyx \prec y if x<yx \lt y in the usual order. That this relation is well-founded is the principle of strong induction.

More generally, let SS be a set of ordinal numbers (or even the proper class of all ordinal numbers), and let xyx \prec y if x<yx \lt y in the usual order. That this relation is well-founded is the principle of transfinite induction.

Similarly, let SS be a set of pure sets (or even the proper class of all pure sets), and let xyx \prec y if xyx \in y. That this relation is well-founded is the axiom of foundation.

Let SS be the set of integers, and let xyx \prec y mean that xx properly divides yy: y/xy/x is an integer other than ±1\pm{1}. This relation is also well-founded, so one can prove properties of integers by induction on their proper divisors.

Revised on March 15, 2014 06:32:55 by Todd Trimble (67.81.95.215)