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Chapter 7: Problem 12

Give an example of a poset with four maximal elements but no greatest element.

Short Answer

Expert verified
An example of poset with four maximal elements but without a greatest element is a set \(P = \{a, b, c, d, e\}\) with relation \(鈮) given by \(a鈮, b鈮, c鈮, d鈮, e鈮, b鈮, c鈮, d鈮\). The maximal elements are \(a, e, b, c\), and there is no greatest element.

Step by step solution

01

Definition

Firstly, understand the terms. A poset (P) is a set with a binary relation (鈮�) satisfying reflexivity (a鈮), antisymmetry (if a鈮 and b鈮 then a=b) and transitivity (if a鈮 and b鈮 then a鈮). An element \(m\) in \(P\) is a maximal element if there is no \(x\) in \(P\) such that \(m\) is strictly less than \(x\), i.e., \(m<x\). An element \(g\) in \(P\) is the greatest element if all \(x\) in \(P\) are less than or equal to \(g\), i.e., \(x鈮\).
02

Construction

Construct a poset with four maximal elements. One simple example would be a set \(P = \{a, b, c, d, e\}\) with the relation \(鈮) defined by \(a鈮, b鈮, c鈮, d鈮, e鈮, b鈮, c鈮, d鈮\). In this case, the maximal elements are \(a, e, b, c\) because there are no elements strictly greater than these in the poset.
03

Verification

Now, verify there is no greatest element. Here, no element has the property that all other elements are less than or equal to it. If the poset had a greatest element, it would mean there is one element that is at least equal to all the others, which is not the case here, thus fulfilling the desired conditions.

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Key Concepts

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

Maximal Elements
In a partially ordered set (poset), a maximal element is an intriguing concept. Imagine a set of elements, each connected by a relation that tells us if one is "less than" or "equal to" another. In this structured set, a maximal element is one that isn't "less than" any other element.

What this means is, for a maximal element \( m \) in a poset \( P \), there is no other element \( x \) such that \( m < x \). In simple terms, you can't find another element above it in the hierarchy.

For example, consider a set \( P = \{a, b, c, d, e\} \) with relations where sometimes \( b \leq a \) and \( c \leq a \). Here, \( a \) is a maximal element as there is no element greater than \( a \). Other maximal elements in this example might include \( e, b, \) and \( c \), if no element exists that is strictly greater than these. This setup confirms that a poset can have several maximal elements.
Greatest Element
A greatest element in a poset is somewhat like a leader; it's the element that stands above all others. If there's a greatest element \( g \) in a poset \( P \), it means every other element \( x \) in that set satisfies the relation \( x \leq g \).

This element is unique because it must be at least as large as every other member of the set. It's important to understand that while every greatest element is maximal, not every maximal element is greatest.

Using the previous example where \( P = \{a, b, c, d, e\} \): if no element emerges that can be identified as greater than all the others, the set lacks a greatest element. In our set, although \( a, e, b, \) and \( c \) are maximal, none is greater than all others, showing no greatest element exists.
Binary Relation in Sets
Binary relations are the backbone of understanding sets in mathematics, especially in posets. A binary relation on a set \( P \) is a way to compare its elements using a rule like "less than or equal to". These comparisons must satisfy three important properties: reflexivity, antisymmetry, and transitivity.

1. **Reflexivity**: Every element \( a \) in \( P \) is related to itself, meaning \( a \leq a \).
2. **Antisymmetry**: If \( a \leq b \) and \( b \leq a \), then \( a = b \). This tells us that distinct elements can鈥檛 be "equal" in terms of relation.
3. **Transitivity**: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \). This property ensures the relation is consistent through the set.

Using these rules, posets create a framework that can accommodate various structures, such as having several maximal elements or sometimes none being the greatest. This understanding helps clarify how relations function within a set.

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Most popular questions from this chapter

a) If \(A=\\{x, y\\}\), how many partial orders on \(A\) have \(x\) as a minimal element? b) For \(B=\\{x, y, z\\}\) how many partial orders on \(B\) have \(x\) as a minimal element?

If the complete graph \(K_{n}\) has 45 edges, what is \(n\) ?

a) Describe the structure of the Hasse diagram for a totally ordered poset \((A, M)\), where \(|A|=n \geq 1\). b) For a set \(A\) where \(|A|=n \geq 1\), how many relations on \(A\) are total orders?

If \((A, \mathscr{})\) is a lattice, with \(A\) finite, prove that \((A, \mathscr{})\) has a greatest element and a least element.

For each of the following relations on the set specified, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive. Also determine whether it is a partial order or an equivalence relation, and, if the latter, describe the partition induced by the relation. a) \(\mathscr{R}\) is the relation on \(\mathbf{Q}\) where \(a \mathscr{b}\) if \(|a-b|<1\). b) Let \(T\) be the set of all triangles in the plane. For \(t_{1}, t_{2} \in T\), define \(t_{1} \mathscr{t}_{2}\) if \(t_{1}, t_{2}\) have the same area. c) For \(T\) as in part (b), define \(\mathscr{R}\) by \(t_{1} \mathscr{t}_{2}\) if at least two sides of \(t_{1}\) are contained within the perimeter of \(t_{2}\). d) Let \(A=\\{1,2,3,4,5,6,7\\}\). Define \(\mathscr{A}\) on \(A\) by \(x \bigcap y\) if \(x y \geq 10\). e) Define \(\mathscr{R}\) on \(\mathbf{Z}\) by \(a \mathscr{G} b\) if \(7 \mid(a-b)\). f) For \(A=\\{1,2,3,4\\} \times\\{1,2,3,4\\}\), define \(\Re\) on \(A\) by \(\left(x_{1}, y_{1}\right) \mathscr{R}\left(x_{2}, y_{2}\right)\) if \(\left(y_{1}-x_{1}\right)=\pm\left(y_{2}-x_{2}\right)\).
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