There's in this relation a distributive lattice?

Question

I need to enlight some points on this exercise. Say me where I wrong and what's the correct answer:

Given $T = \{(2,3),(4,1),(1,1),(7,1),(2,0),(0,4)\}$ and the relation $$(a,b)\rho(c,d)\Leftrightarrow |a-b|\leq|c-d|$$

1. Show the Hasse diagram of $(T, \rho)$
2. Is this a lattice?
3. Is distributive?
4. Is complemented?
5. Has a boolean sub-lattice? If has, show it.

1. This is a chain! $(7,1)$ is the max element, (1,1) the min.
2. Sure a chain has max and min and so is totally ordered, so it's a lattice.
3. I know the properties for a distributive lattice, but how I use in given case?

4. I think it isn't a complemented lattice. Being a chain, for two element $a$ and $b$ at the center of the chain $a \wedge b\neq min(a,b)$ and so $a \vee b\neq max(a,b)$ (where $\wedge$ is the infinum, and $\vee$ the supremum).

   e.g.: $(a,b)=(4,1)$ and $(c,d)=(2,0)$, so $(4,1)\wedge(2,0)$ is not $(1,1)$ (what's the infinum? $(2,0)$ itself or $(2,3)$). Same for supremum. Am I wrong?

   1. If I was right for question 4 (uncomplemented lattice) than the lattice can't be boolean, neither have boolean sublattice

Best regards

Answer 1

You are correct on all counts, except the claim that there's no Boolean sublattice. What about the two element sublattice? Isn't that a complemented distributive lattice?

Also, while I agree the whole lattice is not complemented, I don't understand your explanation of this fact. (If $a\leq b$ then, of course, $a\wedge b = a$. I don't see why you have inequalities in item 4.)

Finally, to answer your question about whether this lattice is distributive, recall that an equivalent condition for a lattice to be distributive is that it has no pentagon or diamond sublattices. That is, a lattice is distributive if and only if it has no $N_5$ sublattices and no $M_3$ sublattices. (See this wikipedia page, which has pictures of $N_5$ and $M_3$.)