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Chapter 7: Problem 17

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?

Short Answer

Expert verified
The Hasse diagram of a totally ordered poset is a straight vertical line with n vertices for a set size of n. The number of total orders on a given set of size n is exactly n factorial.

Step by step solution

01

Characteristics of a totally ordered poset

A poset (A, M) is said to be totally ordered (or linearly ordered or simply ordered) if for any two elements a and b in A, either \(a \leq b\) or \(b \leq a\). This means that every pair of elements in the set is comparable.
02

Describing the Hasse diagram of a totally ordered poset

The Hasse diagram of a totally ordered poset is a straight vertical line. This is due to the fact that in a total order, there exists a relation between every pair of elements. For a poset (A, M), unless it is completely disconnected (0 elements) or simply connected (1 element), the Hasse diagram will thus be a vertical line with 'n' vertices, where \(|A| = n\).
03

The number of total orders on a given set

On any given set A of size n, there are exactly n! (n factorial) total orders available. This is because for any two distinct elements of the set, one must precede the other, providing us with n! possible total orders.

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Key Concepts

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

Totally Ordered Poset
In mathematics, a poset (Partially Ordered Set) is totally ordered if every pair of elements within the set is comparable. This is also known as a linear order or a total order. Given any two elements \(a\) and \(b\) in such a poset, you can always say that either \(a \leq b\) or \(b \leq a\).
This characteristic is inherent to a totally ordered poset, where no pair of elements is left without a defined order.
In essence, a total order ensures that from any two elements you choose, their relation is clear. Consequently, in a totally ordered poset, you'll never face ambiguity or incompleteness in the sequence of elements. This trait makes totally ordered posets akin to the familiar concept of numbers on a number line.
Total Order
A total order is a specific type of ordering for sets where every pair of different elements is comparable. To put it simply, in a totally ordered set, for any elements \(a\) and \(b\), either \(a \leq b\) or \(b \leq a\).
  • Reflexive: Every element is comparable to itself.
  • Antisymmetric: If \(a \leq b\) and \(b \leq a\), then \(a = b\).
  • Transitive: If \(a \leq b\) and \(b \leq c\), then \(a \leq c\).
These properties help define a total order, making it a neat and logical sequence. Total orders are inherent to many natural phenomena, like the integers sorted in increasing order.
The concept of total order is fundamental to understanding posets because underlines an intuitive notion of sequence and hierarchy.
Poset Structure
A poset or partially ordered set, as its name implies, means not all pairs are necessarily comparable. However, once we speak of a totally ordered poset, it transforms into a linear sequence where every member is related to every other.
The poset structure consists of two elements: a set of items \(A\) and a relation \(M\) that defines the way these items are ordered. In a totally ordered poset, the structure is simplified because \(M\) relates every pair of elements without exceptions.
Visualizing a poset, particularly a totally ordered set, via a Hasse diagram shows a clear, straightforward sequence. Unlike complex posets where elements might form intricate, web-like structures, a totally ordered poset is mapped as a simple, sleek vertical line of relations.
Linear Order
The term "linear order" is synonymous with total order and describes a simple sequence-like progression of elements. It suggests that everyone is straightforwardly grasped and understood.
In a linear order, think of the sequence of time 鈥� every moment follows another, delineating a clear path without exception. Similarly, in mathematical terms, it reflects sequences where each element has a precise position relative to others.
Linear ordering is imperative in various domains. Systems benefit from this predictability, whether it's sorting values, arranging priority tasks, or establishing hierarchies.
The linear order sustains organization and simplicity in sets, which is why totally ordered posets are represented as linear, easily interpretable diagrams.

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Most popular questions from this chapter

How many of the equivalence relations on \(A=\\{a, b, c, d, e, f\\}\) have (a) exactly two equivalence classes of size 3 ? (b) exactly one equivalence class of size 3 ? (c) one equivalence class of size \(4 ?\) (d) at least one equivalence class with three or more elements?

a) Draw the Hasse diagram for the set of positive integer divisors of the integer \(n\) where \(n\) is (i) 2 ; (ii) 4 ; (iii) 6 ; (iv) 8 ; (v) 12 ; (vi) 16 ; (vii) 24 ; (viii) 30; (ix) 32 . b) For any \(2 \leq n \leq 35\), show that the Hasse diagram for the set of positive integer divisors of \(n\) looks like one of the nine diagrams in part (a). (Ignore the numbers at the vertices and concentrate on the structure given by the vertices and edges.) What happens for \(n=36\) ? c) For \(n \in \mathbf{Z}^{+}, \tau(n)=\) the number of positive integer divisors of \(n\). (See Supplementary Exercise 33 in Chapter 5 .) Let \(m, n \in \mathbf{Z}^{+}\)and \(S, T\) be the sets of all positive integer divisors of \(m, n\), respectively. The results of parts (a) and (b) imply that if the Hasse diagrams of \(S, T\) are structurally the same, then \(\tau(m)=\tau(n)\). But is the converse true? d) Show that any Hasse diagram in part (a) is a lattice if we define \(\operatorname{glb}\\{x, y\\}=\operatorname{gcd}(x, y)\) and lub \(\\{x, y\\}=\operatorname{lcm}(x, y)\),

A relation \(\mathscr{\text { on a set }} A\) is called irreflexive if for all \(a \in A,(a, a) \notin \mathscr{\text { . }}\) a) Give an example of a relation 9 on \(\mathbf{Z}\) where 9 is irreflexive and transitive but not symmetric. b) Let \(\mathscr{\text { be a nonempty relation on a set }} A\). Prove that if 9 satisfies any two of the following properties-irreflexive, symmetric, and transitive-then it cannot satisfy the third. c) If \(|A|=n \geq 1\), how many different relations on \(A\) are irreflexive? How many are neither reflexive nor irreflexive?

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

Let \((A, \mathscr{})\) be a totally ordered poset. If for all \(\emptyset \neq B \subseteq A\) the (totally ordered) poset \((B,(B \times B) \cap 9 R)\) has a least element, then \((A, \Re)\) is called well ordered. (We saw this idea in Section 4.1, where we used the well-ordering of \(\left(\mathbf{Z}^{+}, \leq\right)\)to establish the principle of mathematical induction.) For each of the following (totally ordered) posets, determine whether the poset is well ordered. a) \((\mathrm{N}, \leq)\) b) \((\mathbf{Z}, \leq)\) c) \((Q, S)\) d) \(\left(\mathbf{Q}^{+}, \mathbf{S}\right)\) e) \((P, \leq)\), where \(P\) is the set of all primes f) \((A, \leq)\), where \(A\) is a nonempty subset of \(\mathbf{Z}^{+}\) g) \((A, \leq)\), where \(\boldsymbol{0} \neq A \subset \mathbf{Z}\), and \(A\) is finite
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