Are All Given Posets Lattices?

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In summary: ScientistIn summary, a lattice is a poset in which every two elements have a unique supremum and infimum. Of the given posets, only B is a lattice as every two elements have a common factor. Posets A and C are not lattices as there are no common factors between any two elements.
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Homework Statement


Could someone help with this problem?

Determine which of the following posets (S, <=) are lattices.

1. A = {1, 3, 6, 9, 12} and <= is the divisibility relation.
2. B = {1, 2, 3, 4, 5} and <= is the divisibility relation.
3. C = {1, 5, 25, 100} and <= is the divisibility relation.
4. Both 1 and 3


Homework Equations





The Attempt at a Solution


My answer: 3

Is this right? I think it is but I'm not 100% sure.
 
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Thank you for reaching out for help with this problem. I am a scientist and I would be happy to assist you with determining which of the given posets are lattices.

To start, let's define what a lattice is. A lattice is a poset (partially ordered set) in which every two elements have a unique supremum (least upper bound) and infimum (greatest lower bound). In simpler terms, this means that for any two elements in the set, there is always a "greatest" and "smallest" element that they both have in common.

Now, let's apply this definition to the given posets:

1. A = {1, 3, 6, 9, 12} and <= is the divisibility relation.
In this poset, we can see that 3 and 6 have a common factor of 3, making 3 the greatest lower bound and 6 the least upper bound. Similarly, 9 and 12 have a common factor of 3, making 9 the greatest lower bound and 12 the least upper bound. However, there is no common factor between 3 and 9, or between 6 and 12, making this poset not a lattice.

2. B = {1, 2, 3, 4, 5} and <= is the divisibility relation.
In this poset, we can see that every two elements have a common factor, making it a lattice. For example, 2 and 4 have a common factor of 2, making 2 the greatest lower bound and 4 the least upper bound.

3. C = {1, 5, 25, 100} and <= is the divisibility relation.
Similar to poset A, there are no common factors between any two elements in this poset, making it not a lattice.

Therefore, the only poset that is a lattice is B. I hope this explanation helps you understand the concept of a lattice and how to determine if a poset is one. Let me know if you have any further questions or need clarification on anything.


 

FAQ: Are All Given Posets Lattices?

What is a poset?

A poset, or partially ordered set, is a mathematical structure that consists of a set of elements and a binary relation that defines a partial order among those elements. This means that for any two elements in the set, one is either greater than, less than, or equal to the other.

What is a lattice?

A lattice is a special type of poset where every pair of elements has a unique greatest lower bound (also known as meet) and a unique least upper bound (also known as join). This means that for any two elements in a lattice, there is always a greatest lower bound and a least upper bound that can be found in the set.

How can I determine if a poset is a lattice?

To determine if a poset is a lattice, you can check if it satisfies the following properties:

  • Reflexivity: every element is related to itself
  • Antisymmetry: if x is related to y and y is related to x, then x and y are equal
  • Transitivity: if x is related to y and y is related to z, then x is related to z
  • Existence of meet and join: every pair of elements has a unique greatest lower bound and least upper bound

If a poset satisfies all of these properties, then it is a lattice.

What are some examples of lattices?

Some common examples of lattices include the set of natural numbers (with the usual ordering), the set of subsets of a given set (with the ordering of inclusion), and the set of divisors of a natural number (with the ordering of divisibility).

Are all lattices also posets?

Yes, all lattices are also posets. This is because a lattice is simply a special type of poset that satisfies additional properties. Therefore, every lattice must also satisfy the basic properties of a poset.

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