Question

(a) Draw the diagram to represent the \(\mid\) (divides) partial order on \\{1,2,3,4,5,6\\} (b) List all the maximal, maximum, minimal, and minimum elements.

Short answer

Maximal: 4, 5, 6. Minimal: 1. Maximum: None. Minimum: 1.

Step-by-step solution

Identify the Partial Order Set

We need to represent the divides partial order (denoted by \(\mid\)) on the set \(\{1, 2, 3, 4, 5, 6\}\). This order means that for any two numbers \(a\) and \(b\) in the set, \(a \mid b\) if \(a\) divides \(b\) without leaving a remainder.

Create the Hasse Diagram Framework

In a Hasse diagram representation of a partial order, there is a line from \(a\) to \(b\) if \(a \mid b\) and there is no \(c\) such that \(a \mid c\) and \(c \mid b\). Start by listing all pairs where a divides b: \(1\mid 2, 1\mid 3, 1\mid 4, 1\mid 5, 1\mid 6, 2\mid 4, 2\mid 6, 3\mid 6.\) Arrange these numbers in a hierarchy based on divisibility.

Construct the Hasse Diagram

- Place 1 at the bottom since it divides all other elements.- 2 and 3 are placed above 1 because 1 \(\mid\) 2 and 1 \(\mid\) 3.- 4 is above 2, because 2 \(\mid\) 4.- 6 is positioned above both 2 and 3 because both 2 \(\mid\) 6 and 3 \(\mid\) 6.- 5 is above 1, as 1 \(\mid\) 5, but not directly above any others because 5 doesn’t divide or isn’t divided by other numbers in the set.
The hierarchy is now visually ordered without lines crossing.

List Maximal and Minimal Elements

- Maximal elements are those with no elements above them. In this diagram, 4, 5, and 6 are the maximal elements. - Minimal elements are those with no elements below them. Here, 1 is the only minimal element.

Identify Maximum and Minimum Elements

- The maximum element is the single element to which no elements in the set are unrelated; however, since there is no single element that every other element divides (like a least common multiple), there is no maximum. - The minimum element is the element which divides all others; thus, 1 is the minimum element, as it divides 2, 3, 4, 5, and 6.

Key concepts

Hasse diagram

A Hasse diagram is a simple and effective way to visualize a partial order relation on a set. In this context, it represents the divisibility relationship among numbers. The diagram consists of points (or nodes) for each element in the set. We draw lines between elements to show the relation, but only if they cannot be linked by intermediate elements. This ensures that the diagram reflects the minimal necessary connections.
For example, if we have the set \(\{1, 2, 4\}\), the Hasse diagram would have:

• A point for each number: 1, 2, and 4.
• A line from 1 to 2, since 1 divides 2 directly.
• Another line from 2 to 4, since 2 divides 4 directly.

This "transitive reduction" property simplifies the understanding of partial orders, making Hasse diagrams a powerful tool for visual representation.

maximal elements

In the context of partial orders, a maximal element is an element that is not smaller than any other element, meaning no other element divides it within the set. There might be more than one maximal element.
Considering the set \(\{1, 2, 3, 4, 5, 6\}\) under the divisibility relation, the maximal elements have no successors. In simpler words, no number divides them aside their multiples within the set. In our specific case, elements 4, 5, and 6 are maximal. None of these elements have direct connections above them in the Hasse diagram, which visualizes that nothing else in the set divides them.
This concept is crucial in areas where determining the boundaries of subsets and understanding the structure of order relations is important.

minimal elements

In a partial order set, minimal elements are those that cannot be reached from any other element. They have no predecessors except themselves. Within our exercise, we only have one minimal element in the set \(\{1, 2, 3, 4, 5, 6\}\) under the divisibility relation: the number 1.
This is because 1 is the only number that can serve as a starting point, dividing all others without being divided by any. In the Hasse diagram, it is placed at the very bottom, illustrating its role in the divisibility hierarchy.
Understanding minimal elements helps establish the base from which all other elements expand in a structured order.

divisibility relation

The divisibility relation is a fundamental concept, especially when dealing with sets and partial orders. It relates two integers if one divides the other with no remainder; symbolically, we say \(a \mid b\) if \(b\) divided by \(a\) equals an integer.
In our set \(\{1, 2, 3, 4, 5, 6\}\), it creates an order by linking numbers where a clear divisible path exists. For instance, 2 divides 4, and 3 divides 6, but 5 doesn't divide any other number in the set.
This relation is often visualized with a Hasse diagram, which clearly outlines these links through direct pathways, without showing transitive steps (like how 1 divides 2 and 2 divides 4, so 1 indirectly divides 4).
Understanding divisibility is key in number theory, with applications ranging from factorization to algorithmic computations and cryptography.