I believe that for subspaces $X,Y,Z$, distributivity does not apply, but could someone give an example to illustrate this: $(X\cap Y)+(X\cap Z)\neq X\cap(Y+Z)$? And perhaps suggest the criteria for this inequality to hold? Thanks.
Non-distributivity of subspaces
-
0see http://mathoverflow.net/questions/17740/is-there-a-version-of-inclusion-exclusion-for-vector-spaces/ – 2011-09-11
2 Answers
Let the vector space be $\mathbb R^2$, $Y$ and $Z$ the $x$- and $y$-axes, respectively, and $X=\{(x,x):x\in\mathbb R\}$.
Assuming you want conditions for the equality to hold: You will find in the (great!) book Quadratic Algebras by Alexander Polishchuk and Leonid Positselski the proof of the following statement
Let $V$ be a vector space and let $W_1$, $\dots$, $W_n$ be subspaces of $V$. Then $W_i$ generate a distributive sublattice in the lattice of all subspaces of $V$ if and only if there exists a basis $\mathcal B$ of $V$ such that each $W_i$ is generated by a subset of $\mathcal B$.
From this, one gets a condition on three subspaces $X$, $Y$, $Z$ of a vector space to generate a distributive lattice. This is probably stronger than what you want, of course. (But it is in fact a theorem that if the equality in your question holds, then it also holds for all permutations of $X$, $Y$, $Z$)
-
0http://en.wikipedia.org/wiki/Lattice_%28order%29 – 2011-09-11
I just happened to come to this page, and I thought the following example may still be of interest to you, even though you posed this question quite a while ago.
Consider the vector space $R^2$. Let $V_1$ and $V_2$ be the subspaces spanned by the standard vectors $\delta_1$ and $\delta_2$ respectively. Then $V_1 + V_2 = R^2$. Next, let $W$ denote the subspace spanned by $\delta_1 + \delta_2$. Then
$W \cap (V_1 + V_2) = W \cap R^2 = W$
But $W \cap V_1 = W \cap V_2 = \{0\}$, the zero subspace of $R^2$, so that
$(W \cap V_1) + (W \cap V_2) = \{0\}.$ That is
$W \cap (V_1 + V_2) \neq (W \cap V_1) + (W \cap V_2)$.
Thus dsitributivity is violated. This is an example from Joseph Jauch, Foundations of Quantum Mechanics, Addison Wesley, 1968. Also see Steven Roman, Advanced Linear Algebra, Springer, 1992
-
0Your example is precisely the same as the one I gave in my answer :P – 2013-08-18