Exercise 17, § 1

A lattice E which has a least element $\alpha$ is said to be relatively complemented if, for each pair of elements $x$, $y$ of E such that $x \leqslant y$, there exists an element $x'$ such that $\sup\,(x, x') = y$ and $\inf\,(x, x') = \alpha$. Such an element $x'$ is called a relative complement of $x$ with respect to $y$.

\* (a) Show that the set E of vector subspaces of a vector space of dimension $\geqslant 2$, ordered by inclusion, is a relatively complemented lattice, but that if $x$, $y$ are two elements of E such that $x \leqslant y$, there exist in general several distinct relative complements of $x$ with respect to $y$. \*

(b) If E is distributive and relatively complemented, show that if $x \leqslant y$ in E, there exists a unique relative complement of $x$ with respect to $y$. E is said to be a Boolean lattice if it is distributive and relatively complemented and if, moreover, it has a greatest element $\omega$. For each $x \in$ E, let $x^*$ be the complement of $x$ with respect to $\omega$. Then the mapping $x \to x^*$ is an isomorphism of E onto the ordered set obtained by endowing E with the opposite ordering, and we have $(x^*)^* = x$. If A is any set, then the set $\mathfrak{P}(\mathrm{A})$ of all subsets of A, ordered by inclusion, is a Boolean lattice.

(c) If E is a complete Boolean lattice (Exercise 11), show that for each family $(x_\lambda)$ of elements of E and each $y \in$ E we have

(Reduce to the case $y = \alpha$, and use the fact that if $\inf\,(z, x_\lambda) = \alpha$ for every index $\lambda$, then $z^* \geqslant x_\lambda$ for every $\lambda$).