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Let $V$ be a vector space over a division ring $D$ and $S$ the set of all subspaces on $V$, partially ordered by set theoretic inclusion.

(i) $S$ is a complete lattice

(ii) $S$ is a complemented lattice; that is for each $V_1\in S$ there exists $V_2\in S$ such that $V=V_1+V_2$ and $V_1\bigcap V_2=0$, so that $V=V_1\oplus V_2$.

(iii) $S$ is a modular lattice; that is, if $V_1,V_2,V_3\in S$ and $V_3\subset V_1$ then $$V_1\cap (V_2+V_3)=(V_1\cap V_2)+V_3.$$

I need help with this problem. I'm not sure where to start. I think I understand what a complete lattice, a complemented lattice and a modular lattice. I'm just not sure how to start. I have ideas but I'm not sure if they are correct. For example I think the LUB subspace is the intersection of all subspaces while the GLB is the vector space V. Any help/hint would be appreciated. I hope somebody could point me in the right direction. Thanks!

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Hints:

  1. The meet of a family is the intersection. The join is the sum.

  2. Take a basis of $V/V_1$ and lift it to $V$.

  3. Take an element of $V_1 \cap (V_2 + V_3)$. That you can write it as both $v_1 \in V_1$, and $v_2 + v_3$ for $v_2 \in V_2$ and $v_3 \in V_3$. That means that $v_2 = v_1 - v_3 \in V_1$.

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  • $\begingroup$ Thanks for the reply. What do I have to show in 1? Do I just have to indicate that if I take any subset of $S$ say $\{V_1,V_2,V_3\}$ Then it has the greatest lower bound $V_1\cap V_2\cap V_3$ and then show that $V_1\cap V_2\cap V_3$ is a subspace also? And then get $V_1+V_2+V_3$ as the least upper bound and show that this is also a subspace? Thank you! $\endgroup$ – chowching Dec 11 '13 at 13:40
  • $\begingroup$ What do you mean by $V/V_1$ is it a quotient or is it the vector space $V$ minus the subspace $V_1$? Thank you. $\endgroup$ – chowching Dec 11 '13 at 13:41
  • $\begingroup$ For the third part I would just show that V_1\cap(V_2+V_3)\subset (V_1\cap V_2)+V_3 and the other way around right? Sorry if I have a lot of questions. Thank you. $\endgroup$ – chowching Dec 11 '13 at 13:45
  • $\begingroup$ For the first part, you have to show that the axioms for a complete lattice are satisfied. For the second, I mean the quotient and not the difference. For the third, take an element of the left-hand side and show that it's an element of the right-hand side. The other inclusion holds for all lattices. $\endgroup$ – arsmath Dec 11 '13 at 14:04
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    $\begingroup$ Let $v_i$ be a basis of $V/V_1$, and treat them as elements of $V$. Consider the subspace they span. $\endgroup$ – arsmath Dec 11 '13 at 14:45

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