Why aren't these lattices isomorphic?

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In summary, the speaker is discussing the isomorphism of two pairs of lattices, one consisting of natural numbers and the other including negative and positive infinity. The speaker initially believed the two pairs to be isomorphic, but then realizes that the presence of 0 in the first pair makes them not isomorphic to the second pair. The expert also clarifies that if the first pair were to include all integers, they would in fact be isomorphic.
  • #1
twoflower
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Hi,

I can't see why these lattices aren't isomorphic:

[tex]
(\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le) \mbox{ and } (\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le)^{inverse}
[/itex]

I thought that this isomorphism would straightforwardly map an element x onto -x in the second lattice, so why aren't these isomorphic please?

I see why these two lattices aren't isomorphic:

[tex]
(\mathbb{N}, \le) \mbox{ and } (\mathbb{N}, \le)^{inverse}
[/itex]

because the element 0 from the first one has no equivalent in the inversed lattice, but the first couple of lattices seems ok to me.




Thank you for clarification.
 
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  • #2
N consists only of positive integers. If you meant Z, all integers, the first pair are isomorphic.
 

1. Why do some lattices have different structures?

Lattices are mathematical structures that can have various configurations, depending on the specific properties of the elements and the operations defined on them. Therefore, it is possible for two lattices to have different structures even if they have the same underlying set of elements.

2. What factors determine the isomorphism of lattices?

The isomorphism of lattices is determined by the relationships between the elements and the operations defined on them. The structure of a lattice depends on its partial ordering, composition, and other properties that dictate its behavior.

3. Are there any lattices that are always isomorphic?

No, there are no lattices that are always isomorphic. Even if two lattices have similar structures, they may differ in certain crucial properties that prevent them from being isomorphic.

4. How can we prove that two lattices are not isomorphic?

To prove that two lattices are not isomorphic, we need to show that they have different properties or structures. This can be done by examining their elements, operations, and relationships between them. If we can find a property that one lattice has but the other does not, then we can conclude that they are not isomorphic.

5. Can we have more than one isomorphism between two lattices?

Yes, it is possible to have multiple isomorphisms between two lattices. This means that there are different ways in which the elements and operations of one lattice can be mapped to those of the other lattice while preserving their structure. However, the number of isomorphisms between two lattices is limited by their properties and structures.

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