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Chapter 7: Problem 7

Give an example of a poset with 5 minimal (maximal) elements but no least (greatest) element.

Short Answer

Expert verified
A possible poset fulfilling the conditions would be the set A = {a1, a2, a3, a4, a5, b, c1, c2, c3, c4, c5} with the partial order '鈮�': for each i from 1 to 5, we have ai 鈮� b and b 鈮� ci. The elements a1, a2, a3, a4, a5 become minimal and c1, c2, c3, c4, c5 become maximal on this poset. However, there is no least or greatest because no single element is comparable to all other elements.

Step by step solution

01

Defining minimal and maximal elements.

In this step, we define minimal elements. They are elements from a set that have no elements smaller than them. The same applies to maximal elements, but in this case, they do not have elements larger than them within the set. This has to happen in a set with an ordered relation that might not be total.
02

Creating the poset

Set A = {a1, a2, a3, a4, a5, b, c1, c2, c3, c4, c5}. We define the partial order '鈮�' for this set as follows: for each i from 1 to 5, ai 鈮� b and b 鈮� ci. This is a partially ordered set (poset) because it fulfills the properties of reflexivity, antisymmetry, and transitivity.
03

Analyzing minimal and maximal elements

Elements a1, a2, a3, a4, a5 are minimal elements, as no other elements in A are less than them. Similarly, elements c1, c2, c3, c4, c5 are maximal elements as there are no other elements in the set A that are greater than them. Although 'b' is greater than each of ai's and less than each of ci's, it is not a greatest or least element because it is not related to all elements in the set.

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