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!



  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 $v_2 = v_1 - v_3 \in V_1$.

| cite | improve this answer | |
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.