length of Lattices

**Goku** · October 4th 2011, 11:28 PM

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...

**emakarov** · October 5th 2011, 11:21 AM

What is the definition of the length of an ordered set? Are we talking about partial or total orders?

**Goku** · October 5th 2011, 01:37 PM

partially ordered sets, where the length is the the size of the longest chain in the poset.

**emakarov** · October 5th 2011, 02:10 PM

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).

Thread: length of Lattices

LinkBack

Thread Tools

Display

length of Lattices

Re: length of Lattices

Re: length of Lattices

Re: length of Lattices

Similar Math Help Forum Discussions

Pairs of lattices

Partial sets/ Least upper bound/Greastest lower bound/Lattices

Packing of Particles in Solids-space Lattices

Length needed from a known length and angle

arc length and parameterized curve length

Search Tags