Partial Order on X: Maximal, Minimal, Greatest & Least Elements

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In summary, a partial order on X is a relation between elements of a set X that is reflexive, anti-symmetric, and transitive. It can be thought of as "less than or equal to" or "precedes" and is commonly used in mathematics. Maximal elements are those that are greater than or equal to all other elements, while minimal elements are those that are less than or equal to all other elements. A greatest element is the unique maximal element, and a least element is the unique minimal element in a partial order on X.
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PeterWatson
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I'm stumbling on this basic question!

Let X = {2,3,4,5,8,9,15,27,45}.
Define a partial order | on X such that x|y <--> x divides y.

(a) Find the maximal and minimal elements
(b) Is there a greatest element, if so what is it?
(c) Is there a least element, if so what is it?
(d) Draw the Hasse diagram

Please help!
 
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Welcome to PF!

Hi Peter! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 

1. What is a partial order on X?

A partial order on X is a mathematical concept that defines a relation between elements of a set X. This relation, denoted by "<=" or "≤", is reflexive, anti-symmetric, and transitive, meaning that for any elements a, b, and c in X, the following conditions must hold true:

  • a ≤ a (reflexivity)
  • If a ≤ b and b ≤ a, then a = b (anti-symmetry)
  • If a ≤ b and b ≤ c, then a ≤ c (transitivity)

This relation can be thought of as "less than or equal to" or "precedes" in some cases, and is commonly used in various areas of mathematics, such as order theory and set theory.

2. What are maximal elements in a partial order on X?

A maximal element in a partial order on X is an element that is greater than or equal to all other elements in X. In other words, there is no other element in X that is strictly greater than the maximal element. A partial order may have multiple maximal elements or none at all.

3. What are minimal elements in a partial order on X?

A minimal element in a partial order on X is an element that is less than or equal to all other elements in X. This means that there is no other element in X that is strictly less than the minimal element. Similar to maximal elements, a partial order may have multiple minimal elements or none at all.

4. What are greatest elements in a partial order on X?

A greatest element in a partial order on X is an element that is greater than or equal to all other elements in X. However, unlike maximal elements, there can only be one greatest element in a partial order. In other words, the greatest element is the unique maximal element in the partial order.

5. What are least elements in a partial order on X?

A least element in a partial order on X is an element that is less than or equal to all other elements in X. Similarly to greatest elements, there can only be one least element in a partial order, which is the unique minimal element in the partial order.

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