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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

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  • $\begingroup$ The title seems to contain a different question than the body of the question. Apart from the fact that the body asks several questions and the title doesn't reflect that, it doesn't even reflect question 3, which is whether the relation is a distributive lattice, not whether it contains one. $\endgroup$ – joriki Mar 29 '12 at 12:15
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    $\begingroup$ Well, you can raise it any time you like by going through your questions and accepting answers :-) $\endgroup$ – joriki Mar 29 '12 at 17:52
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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 $M_3$.)

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  • $\begingroup$ Maybe example wasn't clear. I'll try to explain better so u can say if I wrong. Regardin' the order relation, the chain has this form (from max to min): $(7,1)--(0,4)--(4,1)--(2,0)--(2,3)--(1,1)$. I want to test if lattice is a complemented lattice. To be a c.l., for every couple of elements their infinum needs to be $(1,1)$ and supremum needs to be $(7,1)$. It's right? Now, if I choose $(4,1)$ and $(2,0)$, inf. should be $(2,3)$ and sup. should be $(0,4)$. Is this wrong? $\endgroup$ – BAD_SEED Mar 30 '12 at 10:40
  • $\begingroup$ No. The definition is as follows: a bounded 0-1 lattice $L$ is complemented iff for each $x\in L$ there exists $x' \in L$ such that $x\wedge x' = 0$ and $x\vee x'=1$. Clearly a chain with more than two elements is not complemented, since any element that is not the top nor the bottom has no complement. On the other hand, the two element chain $\langle \{0, 1\}, \wedge, \vee\rangle$ is complemented, since $1$ is the complement of $0$ and vice-versa. $\endgroup$ – William DeMeo Mar 30 '12 at 14:45

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