Let $n$ be a positive integer and let $D(n)$ be the set of positive divisors of $n$. Prove that the distributive lattice $(D(n); |)$ is boolean iff $n$ is square-free.
Let $(D(n),\wedge,\vee,')$ be boolean where the operators are defined by \begin{equation*} x\wedge y=gcd(x,y),\quad x\vee y=lcm(x,y),\quad x'=\dfrac{n}{x}\tag{1} \end{equation*} for each $x,y\in D(n)$.
Since $D(n)$ is boolean, then each $x\in D(n)$ has a complement $x'\in D(n)$ such that $x\wedge x'=gcd(x,x')=1$ and $x\vee x'=lcm(x,x')=n$.
Now $x\wedge x'=gcd(x,x')=1$ implies that $x$ and $x'$ are prime to each other.
Let us consider the pairs of divisors $(x,x')$ of $n$. All such pairs exhaust $D(n)$. Otherwise, if there is any divisor $y$ of $n$ which lies outside the set of pairs, then there must exist $y'\in D(n)$ such that $y\wedge y'=gcd(y,y')\ne 1$ and $y\vee y'=lcm(y,y')\ne n$ which would contradict that $D(n)$ is boolean.
Since $x,x'$ are arbitrary positive divisors of $n$, so the prime factorization of $n$ must be $n=p_1p_2\cdots p_k$ where $p_1,p_2,\cdots,p_k$ are distinct primes.
Consequently, $n$ is square-free.
Conversely, let $n$ be square-free. To show that the distributive lattice $D(n)$ is boolean, it suffices to show that $D(n)$ contains the zero and unit element and each element in $D(n)$ has a unique complement in $D(n)$ where the operators are given by (1).
Let the prime factorization of $n$ be \begin{equation*} n=p_1p_2\cdots p_k \end{equation*} where $p_1,p_2,\cdots,p_k$ are distinct primes.
Now the zero element and unit element of $D(n)$ are respectively $1$ and $n$. Obviously these elements belong to $D(n)$. So $D(n)$ contains the zero and the unit elements.
For any divisor $p_i$, \begin{equation*} p_i'=\dfrac{n}{p_i}=\prod_{\substack{r=1\\r\ne i}}^{k}p_r\end{equation*} and for any divisor $\prod p_i$, \begin{equation*} \left(\prod p_i\right)'=\dfrac{n}{(\prod p_i)}=\prod_{r\ne i} p_r \end{equation*} In each case, $x\wedge x'=1$ and $x\vee x'=n$ for each positive divisor $x$ of $n$, since each $p_i$ is prime.
Thus $D(n)$ is boolean.
$\color{red}{Is\;this\; proof \;correct?}$
