/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(\leqslant\) be a quasi-ord... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 1

Let \(\leqslant\) be a quasi-order on a set \(P\), that is, \(\leqslant\) is reflexive and transitive. Define \(\theta\) on \(P\) by \(a \equiv b(\bmod \theta) \Leftrightarrow a \leqslant b \& b \leqslant a .\) Prove that the relation \(\sqsubseteq\) defined on \(P / \theta\) by $$ [a]_{8} \sqsubseteq[b]_{\theta} \Longleftrightarrow\left(\exists x \in[a]_{\theta}\right)\left(\exists y \in[b]_{\theta}\right) x \leqslant y $$ is an order relation and that \([a]_{\theta} \sqsubseteq[b]_{\theta}\) if and only if \(a \leqslant b\).

Short Answer

Expert verified
The relation \(\sqsubseteq\) is a partial order on \(P/\theta\) equivalent to \(a \leqslant b\) for \([a]_{\theta} \sqsubseteq [b]_{\theta}\).

Step by step solution

01

Define Equivalence Classes

Begin by understanding that \(a \equiv b (\bmod \theta)\) is an equivalence relation on set \(P\). This means for any elements \(a\) and \(b\) in \(P\), they are equivalent if \(a \leqslant b \&\ b \leqslant a\). The equivalence class \([a]_{\theta}\) is the set of all elements in \(P\) equivalently related to \(a\). This forms the partition of \(P\) into disjoint subsets where each element is identified with an equivalence class.
02

Validate Reflexivity

To show that \( \sqsubseteq \) is reflexive, take any equivalence class \([a]_{\theta}\). Pick any element \(x \in [a]_{\theta}\). By reflexivity of \(\leqslant\), \(x \leqslant x\), which means there exist \(x \in [a]_{\theta}\) satisfying \(x \leqslant x\). Thus, \([a]_{\theta} \sqsubseteq [a]_{\theta}\). This ensures \ \sqsubseteq\ is reflexive.
03

Validate Transitivity

To prove transitivity, assume \([a]_{\theta} \sqsubseteq [b]_{\theta}\) and \([b]_{\theta} \sqsubseteq [c]_{\theta}\). This means there exists \(x \in [a]_{\theta}\) and \(y \in [b]_{\theta}\) such that \(x \leqslant y\), and \(y' \in [b]_{\theta}\) and \(z \in [c]_{\theta}\) such that \(y' \leqslant z\). Since \(y\) and \(y'\) are in the same equivalence class, \(y \equiv y'\) implies \(y \leqslant y'\) and \(y' \leqslant y\). Therefore, \(x \leqslant y \leqslant z\), confirming \([a]_{\theta} \sqsubseteq [c]_{\theta}\). Thus, \ \sqsubseteq\ is transitive.
04

Validate Antisymmetry

For antisymmetry, assume \([a]_{\theta} \sqsubseteq [b]_{\theta}\) and \([b]_{\theta} \sqsubseteq [a]_{\theta}\). By definition, there exist \(x \in [a]_{\theta}\) and \(y \in [b]_{\theta}\) such that \(x \leqslant y\), and there exists \(y' \in [b]_{\theta}\) and \(x' \in [a]_{\theta}\) such that \(y' \leqslant x'\). Since \(y\) and \(y'\) are in the same equivalence class, \(y \equiv y'\) implies \(y \leqslant y'\). Together, \(x \leqslant y \leqslant x'\) shows that all elements are equivalent, meeting the condition of antisymmetry: \([a]_{\theta} = [b]_{\theta}\).
05

Conclude that \(\sqsubseteq\) is a Partial Order

Since \(\sqsubseteq\) is reflexive, transitive, and antisymmetric on \(P/\theta\), it establishes that \(\sqsubseteq\) is indeed a partial order relation on the set of equivalence classes.
06

Verify \([a]_{\theta}\) \(\sqsubseteq\) \([b]_{\theta}\) Corresponds to \(a \leqslant b\)

If \(a \leqslant b\) in \(P\), then for any \(x \in [a]_{\theta}\), since \(x \equiv a\), it follows that \(x \leqslant b\). Thus, there exists an \(x \in [a]_{\theta}\) and a \(y = b \in [b]_{\theta}\) such that \(x \leqslant y\). Conversely, if \([a]_{\theta} \sqsubseteq [b]_{\theta}\), then \(a \leqslant b\) by the existence of appropriate \(x\) and \(y\). This confirms the equivalence statement.

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Key Concepts

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

Quasi-order
A quasi-order is a type of binary relation that combines certain properties, making it a broader form of order relation. It's interesting because it encompasses both reflexivity and transitivity, but does not necessarily include antisymmetry.
A relation \( \leqslant \) is a quasi-order if it satisfies the following properties:
  • Reflexivity: For all elements \( a \) in a set \( P \), \( a \leqslant a \).
  • Transitivity: For any elements \( a, b, c \) in the set \( P \), if \( a \leqslant b \) and \( b \leqslant c \), then \( a \leqslant c \).
This means that a quasi-order allows us to express that one thing is "less than or equal to" another while maintaining these specific properties. However, because antisymmetry isn't required, elements can be equivalent in some sense without being identical under a quasi-order, thus lacking the strictness found in partial orders.
Equivalence Relation
Equivalence relations are essential in mathematics for grouping elements based on a defined property such that they exhibit the same behavior or characteristic.
The critical properties of an equivalence relation are:
  • Reflexivity: Each element is related to itself.
  • Symmetry: If \( a \) is related to \( b \), then \( b \) is related to \( a \).
  • Transitivity: If \( a \) is related to \( b \), and \( b \) is related to \( c \), then \( a \) is related to \( c \).
In the context of the problem, the equivalence relation \( a \equiv b (\bmod \theta) \) is established, defining that element \( a \) is equivalent to element \( b \) if they relate under both \( a \leqslant b \) and \( b \leqslant a \), reflecting it as an equivalence class. These classes help partition the set \( P \) into subsets of elements that are indistinguishable in relation to \( \theta \).
Reflexivity
Reflexivity is one of the foundational pillars both in equivalence relations and many order types. It stands for the self-related nature of elements within a set.
Specifically, in a reflexive relation \( \leqslant \), every element is related to itself.
  • This means for any element \( a \), \( a \leqslant a \).
When considering equivalence relations, reflexivity ensures that each item is part of its own equivalence class, which justifies the creation of these classes uniformly.
In the problem, reflexivity guarantees that a relation applied to any equivalence class \([a]_{\theta}\) reflects self-consistency, ensuring that \([a]_{\theta} \sqsubseteq [a]_{\theta}\), making reflexivity a crucial aspect of establishing the partial order \( \sqsubseteq \) on \( P/\theta \).
Transitivity
Transitivity plays a pivotal role in linking chains of relations across elements. It ensures that indirect relationships become direct.
In an order relation, transitivity requires that if an element \( a \) is related to \( b \), and \( b \) is related to \( c \), then \( a \) must be related to \( c \).
  • This entails that solutions to chains of relations can be collapsed into a direct connection, such as if \( a \leqslant b \) and \( b \leqslant c \), then necessarily, \( a \leqslant c \).
In equivalence and order relations, transitivity ensures smooth and logical progression through connected elements.
For the given context, transitive property ensures that the partial order \( \sqsubseteq \) enables sustained connections across equivalence classes, validating further aggregate relationships like \([a]_{\theta} \sqsubseteq [b]_{\theta} \) and \([b]_{\theta} \sqsubseteq [c]_{\theta} \), so that \([a]_{\theta} \sqsubseteq [c]_{\theta} \).
Antisymmetry
Antisymmetry is essential for defining the uniqueness of order relations, such as a partial order, distinguishing them from equivalence relations.
In an antisymmetric relation, if two distinct elements \( a \) and \( b \) are mutually related, then \( a \) and \( b \) must be identical.
  • This condition is formally expressed as: If \( a \leqslant b \) and \( b \leqslant a \), then \( a = b \).
This property ensures that while elements can overlap in order comparison, such overlap confirms their sameness, preventing any arbitrary convergence.
For the relation \( \sqsubseteq \) in the exercise, antisymmetry confirms that two equivalence classes \([a]_{\theta}\) and \([b]_{\theta}\) are actually the same class if they are related in both directions, thus confirming \([a]_{\theta} = [b]_{\theta}\). This characteristic solidifies \( ^{\backslash}{^t} \sqsubseteq \) as not just any relation but specifically as a partial order.

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Most popular questions from this chapter

Show that a homomorphism \(f: L \rightarrow K\) is an embedding if and only if ker \(f=0\)

(i) Let \(L, K_{1}\) and \(K_{2}\) be lattices and assume that the map \(\varphi: L \rightarrow K_{1} \times K_{2}\) is an embedding. Define congruences \(\theta_{1}, \theta_{2} \in \operatorname{Con} L\) by \(\theta_{i}=\operatorname{ser}\left(\pi_{i} \circ \varphi\right)\) where \(\pi_{i}: K_{1} \times K_{2} \rightarrow K_{i}\) is the natural projection \((i=1,2)\). Show that \(\theta_{1} \wedge \theta_{2}=0\) in Con \(L\). (ii) Let \(\theta_{1}, \theta_{2} \in \operatorname{Con} L\) with \(\theta_{1} \wedge \theta_{2}=0\). Show that the map \(\psi: L \rightarrow L / \theta_{1} \times L / \theta_{2}\) defined by \(\psi(a):=\left([a]_{\theta_{1}},[a]_{\theta_{2}}\right)\) is an embedding. (iii) Use (i) to show that each of the following lattices \(L\) has the property that if \(L \mapsto K_{1} \times K_{2}\), then \(L \mapsto K_{1}\) or \(L \mapsto K_{2}\) (a) \(L=\mathbf{M}_{3}\) (b) \(L=\mathbf{N}_{5}\) (c) \(L=\mathbf{M}_{3,3}\) (see Figure 4.5). [Hint. If you haven't already done so, you'll first need to calculate Con \(L\) in each case (Exercises \(6.11\) and \(6.13\) ). Then use Exercise 6.3.]

Let \(L\) be a distributive lattice and assume that \(c \leqslant d\) in \(L .\) Prove that \((a, b) \in \theta(c, d) \Leftrightarrow a \wedge c=b \wedge c \& a \vee d=b \vee d\).

Let \(J\) be an ideal of a lattice \(L\) and define a relation \(\theta_{J}\) on \(L\) by $$ \theta_{J}=\left\\{(a, b) \in L^{2} \mid(\exists c \in J) a \vee c=b \vee c\right\\} $$ Prove that \(L\) is distributive if and only if for every ideal \(J\) of \(L_{1}\) the relation \(\theta_{J}\) is a congruence on \(L\) and \(J\) is a block of the corresponding partition of \(L\).
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