1. ## 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???

2. ## 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...

3. ## 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...

4. ## Re: Pairs of lattices

You are right. If the maximum element $a$ of M covers $a_1,\dots,a_m$ and the maximum element $b$ of N covers $b_1,\dots,b_n$, then $(a,b)$ covers $(a_1,b),\dots,(a_m,b)$, $(a,b_1),\dots,(a,b_n)$. 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.