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I was assigned a Homework question that says this

Prove that every non-empty interval in a partition lattice either has an order isomorphism to a partition lattice or an order-isomorphism to a cartesian product of $n$ partition lattices for some $n > 1$.

How can I go about starting to answer this question?

Zygormatic
  • 3
  • Look at partition blocks of the interval’s bounds, for example. – arseniiv Dec 13 '17 at 23:22
  • So, would something similar to something that I found on this site would work, IE: https://math.stackexchange.com/questions/1697791/isomorphism-of-power-set-and-cartesian-product

    or is this something completely different?

     – Zygormatic Dec 13 '17 at 23:59
  • That’s pretty unrelated. You have lattice isomorphism, there is ring isomorphism; not the same thing. :) – arseniiv Dec 14 '17 at 00:04
  • Say that If we created a Hasse Diagram of a partition lattice for partitions of a four element set, then we could find an isomorphism from a non-empty interval of the same partition lattice? – Zygormatic Dec 14 '17 at 00:30
  • From a proper sublattice to the full lattice, if it’s finite—of course no. But generally a wish to look at Hasse diagram is good. Draw some intervals in there, consider what they look like. Then maybe return to my first comment: what blocks are in the lower bound of the interval; what are in the upper bound; is there some mapping? – arseniiv Dec 14 '17 at 00:43
  • So when I created my hasse diagram for the partitons of the set $({1,2,3,4} ,>=)$ and refined it, I got a straight line, by looking at the definition of a upper bound "Let S ⊆ A in the poset $(A, >=)$. If there exists an element u ∈ A such that s 4 u for all s ∈ S, then u is called an upper bound of S." then my upper bound should be 4? or am I misunderstanding something. – Zygormatic Dec 14 '17 at 00:57
  • Honestly, I haven’t understood the previous. There are some examples for us to be on the same page. Here is a Hasse diagram with an interval $[{{1},{2},{3},{4}};; {{1,2},{3,4}}]$ depicted, and here is an interval $[{{1,2},{3},{4}};; {{1,2,3,4}}]$ (bounds may be interchanged depending on your definition of order in a partition lattice). – arseniiv Dec 14 '17 at 01:34
  • Ok, I understand that, but the diagram is created from {1,2,3,4} is that the same as what you have depicted above? and If so, does the portion you posted say that the diagram of {1,2,3,4} is isomorphic because its bounds could be matched with each other?

    I'm sorry if I seem to not know what you are telling me, we haven't talked about isomorphism in partition lattices yet.

     – Zygormatic Dec 14 '17 at 02:26
  • Hm, and about lattices in generall too? Isomorphism here should not be between the full partition lattice (of the set ${1,2,3,4}$) and something, but between a sublattice (in images above, these sublattices are highlighted) and some other, smaller, partition lattice (or a cartesian product thereof). Look what blocks of a finer interval bound refine what blocks of its coarser bound, they literally sell it all out. – arseniiv Dec 14 '17 at 03:19
  • Ok, thank you.I just needed help clearing up what is what, I appreciate your help and all you have done. – Zygormatic Dec 14 '17 at 03:42

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