length of Lattices

• Oct 4th 2011, 11:28 PM
Goku
length of Lattices
Prove that

length(P x Q) = length(P) + length(Q)

where P, Q are ordered sets of finite length.

Any ideas of where to start...
• Oct 5th 2011, 11:21 AM
emakarov
Re: length of Lattices
What is the definition of the length of an ordered set? Are we talking about partial or total orders?
• Oct 5th 2011, 01:37 PM
Goku
Re: length of Lattices
partially ordered sets, where the length is the the size of the longest chain in the poset.
• Oct 5th 2011, 02:10 PM
emakarov
Re: length of Lattices
If $a_1,\dots,a_n$ and $b_1,\dots,b_m$ are chains in P and Q, respectively, then $(a_1,b_1),(a_2,b_1),\dots,(a_n,b_1),$ $(a_n,b_2),\dots,(a_n,b_m)$ is a chain in P x Q, so length(P x Q) >= length(P) + length(Q). Conversely, if you have a chain $(a_1,b_1)>\dots>(a_n,b_n)$ in P x Q, then for each $i$ you have $a_{i+1}>a_i$ or $b_{i+1}>b_i$ (or both). So, you can construct chains in P and Q whose total length is >= n. This means that length(P) + length(Q) >= length(P x Q).