Mark the following statements true $(T)$ or false $(F)$ justifying your answer briefly.
a) Let $D$ be an ordered integral domain with positive subset $P$. In $D[x]$, we write
$$
\left(a_0+a_1 x+\cdots+a_n x^n\right) \in P^*, a_n \neq 0
$$
if, and only if, $a_n \in P$ in $D$. Then, $D[x]$ with positive set $P^*$ is an ordered integral domain.
b) $\mathbb{Z}[\sqrt{3}]=\{a+b \sqrt{3}: a, b \in \mathbb{Z}\}$ can be made an ordered integral domain, by defining the set $P$ of positive elements.
c) Let $n=p_1^{a_1}, p_2^{a_2}, \ldots, p_k^{a_k}\left(p_1<p_2 \cdots<p_k\right)\left(p_i\right.$, primes, $\left.i=1,2, \ldots, k\right)$ $D(n)$, the set of divisors of $n$ partially ordered by divisibility forms a distributive lattice $L_1 \times L_2 \times \cdots L_k$ where $L_i$ is a chain of length $a_i(i=1,2, \ldots, k)$.
d) There exists a modular lattice of 7 elements in which the complemented elements do not form a sublattice.
e) Any Boolean algebra generated by $n$ elements has more than $2^{2^n}$ elements.
f) The modular lattice of all subspaces of a finite dimensional vector space $V$ is also a complemented lattice.