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Chapter 8: Problem 22

Let \(P\) be an ordered set and \(Q\) a complete lattice. Given a map \(f: P \rightarrow Q\), define \(\bar{f}: P \rightarrow Q\) by $$ \bar{f}(x)=\bigvee\\{f(y) \mid y \leqslant x\\} $$ Show that \(\bar{f}\) is order-preserving and that \(f=\bar{f}\) if and only if \(f\) is order-preserving. Show further that \(F\) defined by \(F: f \mapsto \bar{f}\) is an order-preserving map from \(Q^{P}\) to \(Q^{(P)} \subseteq Q^{P}\) whose set of fixpoints is exactly \(Q^{(P\rangle}\).

Short Answer

Expert verified
\(\bar{f}\) is order-preserving. \(f = \bar{f}\) if and only if \(f\) is order-preserving. \(F\) is an order-preserving map from \(Q^P\) to \(Q^{(P)}\) with fixpoints in \(Q^{(P\rangle}\).

Step by step solution

01

Define Order-Preserving

A map \(f: P \rightarrow Q\) is said to be order-preserving if for all \(x, y \in P\) with \(x \leqslant y\), we have \(f(x) \leqslant f(y)\) in \(Q\). Here, our goal is to prove that \(\bar{f}\) as defined is an order-preserving operator.
02

Show \(\bar{f}\) is Order-Preserving

For any \(x, y \in P\) with \(x \leqslant y\), consider \(\bar{f}(x) = \bigvee \{ f(z) \mid z \leqslant x \}\) and \(\bar{f}(y) = \bigvee \{ f(z) \mid z \leqslant y \}\). Since \(x \leqslant y\), the set \(\{z \mid z \leqslant x\}\) is a subset of \(\{z \mid z \leqslant y\}\), implying that \(\bar{f}(x) \leqslant \bar{f}(y)\). Hence, \(\bar{f}\) is order-preserving.
03

Show \(f = \bar{f}\) if and only if \(f\) is Order-Preserving

If \(f = \bar{f}\), for each \(x \in P\), \(f(x) = \bigvee \{f(y) \mid y \leqslant x\}\). For any \(x \leqslant y\), this implies \(f(x) \leqslant f(y)\). Conversely, if \(f\) is order-preserving, then \(f(x) = \bigvee \{f(y) \mid y \leqslant x\}\), hence \(f = \bar{f}\).
04

Show \(F: f \mapsto \bar{f}\) is Order-Preserving

To show that \(F: Q^P \rightarrow Q^{(P)} \subseteq Q^P\) is order-preserving, assume \(f \leq g\) for \(f, g: P \rightarrow Q\). Then for any \(x \in P\), \(f(y) \leq g(y)\) for all \(y\), implies \(\bigvee \{f(y) \mid y \leq x\} \leq \bigvee \{g(y) \mid y \leq x\}\), showing \(\bar{f}(x) \leq \bar{g}(x)\), hence \(F(f) \leq F(g)\).
05

Define the Fixpoints of \(F\)

A map \(f\) is a fixpoint of \(F\) if \(F(f) = f\), which means, according to our earlier result, that \(f = \bar{f}\). This occurs if and only if \(f\) is order-preserving, defining the fixpoints as precisely \(Q^{(P\rangle}\), the subset of order-preserving maps from \(P\) to \(Q\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complete Lattice
In mathematics, a complete lattice is a special type of ordered set where every subset has both a supremum (also known as the join or least upper bound) and an infimum (greatest lower bound). This property makes complete lattices highly structured and allows for robust manipulation. To visualize this, imagine a collection of numbers drawn from a wider set; a complete lattice guarantees there's a smallest number (infimum) and a largest number (supremum) within that collection.
  • An important property of complete lattices is that they include the ability to "inflate" sets to include all their limits, providing a way to talk about boundaries and extremities.
  • This characteristic is crucial when dealing with concepts like fixpoints and ordered sets because it ensures that any limited element collections can reach their maximum boundary.
  • Applications include computer science data structures, decision-making models, and more, where a defined end-point or boundary helps model and control systems.
Understanding the complete lattice concept contributes to recognizing more complex abstract structures in mathematics and related fields.
Order-Preserving Map
An order-preserving map, sometimes known as a monotone function, is a type of function between two ordered sets that maintains the order relationship. Formally, if we have two elements, say \( x \) and \( y \), in set \( P \) such that \( x \leq y \), an order-preserving map \( f: P \rightarrow Q \) ensures that their images in \( Q \) follow the same order, so \( f(x) \leq f(y) \).
  • Such maps are foundational in mathematics and appear in various domains like statistics, economics, and computer science as they help maintain structure across transformations.
  • They support the creation of more complex mathematical objects, ensuring transformations do not disrupt the inherent order that might be critical for a system's function.
  • Order-preserving maps help translate real-world scenarios into functions that reflect relative levels or ranks, making them pivotal in modeling agreements, preferences, and strategies.
Understanding order-preserving maps assists in grasping how mathematical structures can be reliably manipulated and analyzed without altering essential relationships.
Fixpoints
Fixpoints play a pivotal role in mathematics and other scientific fields. A fixpoint of a function is an element that is mapped to itself by the function. More formally, if we have a function \( f \), a point \( x \) is a fixpoint if \( f(x) = x \). This concept is particularly important in the context of a map like \( F \), which is defined by sending \( f \) to \( \bar{f} \).
  • Such fixpoints are integral in studies involving equilibrium states, such as solving differential equations where the solution does not change upon further application of the function.
  • The existence of fixpoints can be used to prove stability and convergence within a system, offering assurances that certain operations will lead to steady states or unchanging conditions.
  • In our context, a fixpoint is significant for determining when \( f \) and \( \bar{f} \) align perfectly, indicating an inherent preservation of order.
Grasping the concept of fixpoints enables one to explore deeper ideas in both finite and infinite context scenarios while understanding the continuity and steadiness inherent in the system described.
Map from P to Q
A map from set \( P \) to set \( Q \), denoted as \( f: P \rightarrow Q \), symbolizes a function assigning each element in \( P \) exactly one element in \( Q \). This concept is the cornerstone of the exercise as it builds on simple assignments to explore complex structures.
  • In the given problem, maps serve as the initial step to discuss transformations involving order and lattices, setting up the stage to explore more intricate relationships.
  • The idea of mapping is used widely in practically all disciplines, translating inputs into outputs in predictable patterns, and is the center of subjects like calculus, topology, and functional analysis.
  • In optimization, mappings translate decision variables to corresponding outcomes, thus being integral in designing algorithms that must account for all possible scenarios reflecting changes and impacts.
Understanding how mappings operate from \( P \) to \( Q \) forms the rudimentary layer upon which the exercise's complexities are built, allowing for meaningful interactions and insights to emerge.

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Most popular questions from this chapter

Let \(P_{1}\) and \(P_{2}\) be CPOs. Let \(D \subseteq P_{1} \times P_{2}\) be a directed set. Define \(D_{1}\) and \(D_{2}\) by $$ \begin{aligned} &D_{1}:=\left\\{x_{1} \in P_{1} \mid\left(\exists x_{2} \in P_{2}\right)\left(x_{1}, x_{2}\right) \in D\right\\} \\ &D_{2}:=\left\\{x_{2} \in P_{2} \mid\left(\exists x_{1} \in P_{1}\right)\left(x_{1}, x_{2}\right) \in D\right\\} \end{aligned} $$ Show that \(D_{1}\) and \(D_{2}\) are directed and that \(\bigcup D=\left(\sqcup D_{1}, \bigcup D_{2}\right)\). Deduce that \(P_{1} \times P_{2}\) is a CPO.

Let \(P\) be a CPO and \(F\) an order-preserving self-map on \(P .\) Let $$ \begin{aligned} Q &:=\operatorname{post}(F) \cap \mathrm{fix}(F)^{\ell} \\ &=\\{x \in P \mid x \leqslant F(x) \&(\forall y \in \operatorname{fix}(F)) x \leqslant y\\} \end{aligned} $$ (i) Show that \(Q\) is a sub-CPO of \(P\). (ii) Show that if \(Q\) has a maximal element, \(\beta\), then the least fixpoint \(\mu(F)\) of \(F\) exists and equals \(\beta\). (It is important to notice that in the definition of \(Q\) we are not claiming that \(F\) has a fixpoint. The second clause in the definition of \(Q\) is satisfied vacuously if \(f_{x}(F)=\varnothing .\) It serves to ensure that a fixpoint of \(F\) in \(Q\) is the least fixpoint of \(F\).)

Given \(f: \mathbb{N}_{0} \rightarrow\left(\mathbb{N}_{0}\right)_{\perp}\), let \(\bar{f}\) be defined by $$ \bar{f}(k)=\left\\{\begin{array}{ll} 1 & \text { if } k=0 \\ k f(k-1) & \text { if } k \geqslant 1 \text { and } f(k-1) \neq 1 \\ \perp & \text { otherwise } \end{array}\right. $$ Define \(F:\left(\mathbb{N}_{0} \rightarrow\left(\mathbb{N}_{0}\right)_{\perp}\right) \rightarrow\left(\mathbb{N}_{0} \rightarrow\left(\mathbb{N}_{0}\right)_{\perp}\right)\) by \(F(f)=\bar{f}\). (i) Show that \(F\) is order-preserving: (ii) Prove that \(F\) is continuous. [Hint. Let \(D=\left\\{g_{i}\right\\}_{i \in I}\) be a directed set in \(\left(\mathrm{N}_{0} \rightarrow\left(\mathbb{N}_{0}\right)_{\perp}\right)\), and let \(g:=\bigcup_{i \in I} g_{i} .\) Check that \(\bar{g}(k)=\bigcup_{i \in I} \bar{g}_{i}(k)\) for all \(k \in \mathbb{N}_{0}\) by considering the cases (a) \(k=0\) (b) \(g_{i}(k-1)=\perp\) for all \(i \in I\) (c) \(g_{i}(k-1) \neq \perp\) for some \(\left.i \in I .\right]\) (This exercise establishes the continuity of the map of which the factorial function is the least fixpoint.)

Construct the least solutions to the following equations in the CPO \(P=\wp(X)\) (i) \(S=S \cup T \quad(T\) fixed \()\) (ii) (with \(X=\mathbb{N}) S=S \cup\\{1\\} \cup\\{n+2 \mid n \in S\\}\) (iii) (with \(X=\mathbb{N}^{2}\) ) $$ S=\\{(n, n) \mid n \in \mathbb{N}\\} \cup\\{(n, m+1) \mid(n, m) \in S\\} $$ (iv) (with \(X\) a finite group) \(S=T \cup S \cup S \cdot S \quad(T\) fixed). [Hint. Find a suitable order-preserving map \(F\) of which the required set is to be the least fixpoint, guess a formula for \(F^{n}(\perp)\) and prove by induction that it works, and finally verify that \(\bigcup F^{n}(\perp)\) is a fixpoint.] (See Theorem \(8.15(\mathrm{i}) .)\)

Let \(L\) be a complete lattice and define \(F: \wp(L) \rightarrow \wp(L)\) by \(A \mapsto \downarrow \bigvee A\). Show that \(F\) is order-preserving and that \(f x(F) \cong L\) (This is the long proof that every complete lattice is, up to isomorphism, a lattice of fixpoints. Find the short proof.)
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