# Pairs of lattices

• Sep 28th 2011, 12:16 PM
Goku
Pairs of lattices
Find all pairs of lattices (L,K)(up to isomorphism) such that

a) L x K contains 20 elements
b) L is non-distributive
c) K has at least 3 elements
d) the greatest element of L x K covers precisely 4 elements.

From what I understand from this question K can have 4, 5, 10, 20
elements, but since L is non-distributive it has to be at least 5 elements since it has to be isomorphic to M3 and N5.
So K has to be 4 elements.

How do you find these lattices???
• Sep 29th 2011, 02:50 AM
emakarov
Re: Pairs of lattices
By some element x covering four other elements I am assuming the question means that there are four elements immediately under x, with no intermediate elements. Is it true that if greatest element of one lattice covers m elements and the greatest element of another covers n elements, then the greatest element of the product covers mn elements? If so, then L cannot be M3...
• Sep 29th 2011, 07:39 AM
Goku
Re: Pairs of lattices
I do not think that it will cover mn elements (this is just a guess)...

Take L= M3 and M to be the chain 4 this L x K covers 4 elements... maybe it covers m+n elements...

Any ideas...
• Sep 29th 2011, 12:45 PM
emakarov
Re: Pairs of lattices
You are right. If the maximum element \$\displaystyle a\$ of M covers \$\displaystyle a_1,\dots,a_m\$ and the maximum element \$\displaystyle b\$ of N covers \$\displaystyle b_1,\dots,b_n\$, then \$\displaystyle (a,b)\$ covers \$\displaystyle (a_1,b),\dots,(a_m,b)\$, \$\displaystyle (a,b_1),\dots,(a,b_n)\$. So, either L is M3 and the top element of K has 1 descendant, ot L is N5 and the maximum of K has 2 descendants.