Definition 6.1 a Binary Relation R on a Set X Is Called 1. Reflexive If Xrx For

Home , Binary relation, Identity function

Deﬁnition 6.1 A binary relation R on a set X is called 1. reﬂexive if xRx for all x in X 2. anti-symmetric if, whenever xRy and yRx, it follows that x = y 3. transitive if, whenever xRy and yRz, it follows that xRz.

Deﬁnition 6.2 A partial order is a set P together with a reﬂexive, anti-symmetric, and transitive binary relation on it.

The binary relation in a partial order is typically written as ≤.

Example 6.3 Typical examples of partial orders are as follows: 1. the set of integers Int together with the usual ordering of integers 2. the set of integers Int with the ordering given by equality 3. the set P(X) of subsets of a set X, with ordering determined by inclusion, i.e., A ≤ B precisely when A ⊆ B.

We now proceed to the example of a partial order of primary importance to us in these notes.

Deﬁnition 6.4 Given sets X and Y , a partial function from X to Y is given by a subset X0 of X together with a function f : X0 −→ Y . We normally write the partial function (X0, f) simply as f. The set X0 is called the domain of the partial function f and is written dom(f). We denote the set of partial functions from X to Y by X*Y (note that there is only half an arrowhead in X*Y ).

So every function from X to Y can be seen as a partial function with domain X0 = X. There is also a partial function with empty domain, i.e., with X0 the empty subset of X. A common example of a partial function is that from the set of natural numbers Nat to itself that sends n to n − 1 if n ≥ 1 but is undeﬁned if n = 0. We can now describe our leading example of a partial order:

Example 6.5 Given partial functions f, g : X*Y , put f ≤ g if 1. dom(f) ⊆ dom(g) and 2. whenever f(x) is defined (and therefore g(x) is also defined), f(x) = g(x). It is routine to check that ≤ satisfies the axioms required to make it a partial order, i.e., that ≤ is reflexive, anti-symmetric, and transitive. For instance, for reflexivity, given a partial function from X to Y , i.e., given a subset X0 of X and a function f : X0 −→ Y , it is trivially the case that dom(f) = dom(f), and that, whenever f(x) is defined, f(x) = f(x), and so, by our definition, f ≤ f. This partial order is called the partial order of partial functions from X to Y .

16 For a speciﬁc example of this, let X = Y be the set Nat of natural numbers. Let f send all even numbers 2n to 2n + 1 but be undeﬁned on odd numbers. And let g send all numbers m to m + 1. Then f is less than or equal to g in the partial order given by Example 6.5.

Deﬁnition 6.6 Given partial orders (P, ≤P ) and (Q, ≤Q), a function h : P −→ Q is called order preserving if h preserves the partial order, i.e., if x ≤P y implies that h(x) ≤Q h(y).

For an example, consider the discrete partial order on the natural numbers, i.e., n is put less than or equal to m if and only if n = m. Call this partial order N. Now consider the ordinary partial order on the natural numbers: call it N 0. Then the identity map from N to N 0 is order preserving.

Definition 6.7 Given an order preserving function h : P −→ P from a partially ordered set P to itself: • a fixed point for h is an element x of P for which h(x) = x • a least fixed point for h is a fixed point lfp(h) of h for which, for any fixed point F of h, one has lfp(h) ≤ F .

An order preserving function might not have any fixed points: for example, the function from Int to Int, with the usual partial order, that sends an integer n to n + 1 has no fixed points. Or it might have many fixed points, for instance the identity function on Int. And if it does have a fixed point, there is not necessarily a least one, e.g., for identity maps as above. But if there does exist a least fixed point, it is necessarily unique. In order to give denotational semantics for while, we shall use the least fixed point of an order preserving function on the partial order of partial maps Σ * Σ.

Deﬁnition 6.8 Given a partial order (P, ≤) and a subset A of P , a least upper bound of A is an element lub(A) of P such that • a ≤ lub(A) for every a ∈ A and • for any element c such that a ≤ c for all a ∈ A, it follows that lub(A) ≤ c.

Deﬁnition 6.9 1. ω denotes the partially ordered set given by the set of natural numbers with the usual ordering, cf 6.3.

2. An ω-chain in a poset P is an order preserving function c : ω −→ P , i.e., a sequence of elements of P of the form c0 ≤ c1 ≤ c2 ≤ .... (In this setting, c(n) is usually written cn.)

3. A poset P is ω-complete if every ω-chain c0 ≤ c1 ≤ c2 ≤ ... in P has a least upper bound cω.

17 Theorem 6.10 For any sets X and Y , the partial order of partial functions from X to Y has a least element and is ω-complete.

Proof The least element of X*Y is given by the partial function with empty domain. For ω-completeness, suppose we are given an ω-chain f0 ≤ f1 ≤ f2 ≤ ... in X*Y . We must find a partial function h : X*Y that is a least upper bound for it. Let H be the subset of X given by the union over all n ∈ ω of dom(fn). So, for any x ∈ H, there must be at least one n ∈ ω such that x ∈ dom(fn). Choose one such n and define h(x) = fn(x). For any n and m, either fn ≤ fm or fm ≤ fn. So either dom(fn) ⊆ dom(fm) or dom(fm) ⊆ dom(fn), and the values of fn(x) and fm(x) agree on their common domain. So for any x, any choice of n will yield the same definition of h(x). So, for every n ∈ ω, we have dom(fn) ⊆ dom(h) and, for every x ∈ dom(fn), we have fn(x) = h(x). So fn ≤ h, making h an upper bound. It remains to check that h is the least upper bound. For that, suppose k is a partial function from X to Y and fn ≤ k for each n. So dom(fn) ⊆ dom(k) and for each x in dom(fn), we have fn(x) = k(x). But dom(h) is the union of the dom(fn)’s, so dom(h) is a subset of dom(k). Any element x of dom(h) is an element of dom(fn) for some n, and so h(x) = fn(x) = k(x). Thus h ≤ k.

Deﬁnition 6.11 Given ω-complete partial orders (P, ≤P ) and (Q, ≤Q), an order preserving function h : P −→ Q is called ω-continuous if h preserves least upper bounds of ω-chains, i.e., for any ω-chain c0 ≤ c1 ≤ c2 ≤ ... in P , one has h(cω) is the least upper bound of h(c0) ≤ h(c1) ≤ h(c2) ≤ ....

Proposition 6.12 Given an ω-complete partial order P with a least element, every ω-continuous function h : P −→ P has a least ﬁxed point, which we denote by fix(h).

Proof Define h0 to be the least element 0 of P . Then inductively put hn+1 = h(hn). By induction and because h preserves the partial order, it follows, for every n, that hn ≤ hn+1. So the set {h0, h1,...} forms an ω-chain. So h preserves its least upper bound hω. Thus h(hω) is the least upper bound of h(h0) ≤ h(h1) ≤ h(h2) ≤ ..., but by construction, that chain is precisely h1 ≤ h2 ≤ h3 ≤ ..., which has least upper bound hω. So h(hω) = hω, making hω a fixed point of h. It remains to check that hω is the least fixed point. Suppose h(x) = x. We defined h0 to be the least element 0 of P , so h0 ≤ x. In order to apply induction, assume hn ≤ x. Then, since h is order preserving, h(hn) ≤ h(x). But hn+1 = h(hn) and h(x) = x, so hn+1 ≤ x. Thus, by induction, for all n, we have hn ≤ x. So the least upper bound hω is less than or equal to x, completing the proof.