%
\label{prop:distributive-lattice}
The triple $(\tau\text{-Idx}, \cap_\tau, \cup_\tau)$
forms a \textbf{distributive lattice}.
In particular, the following identities hold
for all $\underline{a}, \underline{b}, \underline{c} \in \tau\text{-Idx}$:
\begin{enumerate}
    \item \textbf{Commutativity:}
          \[
              \underline{a} \cup_\tau \underline{b}
              = \underline{b} \cup_\tau \underline{a},
              \quad
              \underline{a} \cap_\tau \underline{b}
              = \underline{b} \cap_\tau \underline{a}.
          \]

    \item \textbf{Associativity:}
          \[
              (\underline{a} \cup_\tau \underline{b}) \cup_\tau \underline{c}
              = \underline{a} \cup_\tau (\underline{b} \cup_\tau \underline{c}),
          \]
          \[
              (\underline{a} \cap_\tau \underline{b}) \cap_\tau \underline{c}
              = \underline{a} \cap_\tau (\underline{b} \cap_\tau \underline{c}).
          \]

    \item \textbf{Idempotence:}
          \[
              \underline{a} \cup_\tau \underline{a}
              = \underline{a},
              \quad
              \underline{a} \cap_\tau \underline{a}
              = \underline{a}.
          \]

    \item \textbf{Absorption:}
          \[
              \underline{a} \cup_\tau (\underline{a} \cap_\tau \underline{b})
              = \underline{a},
              \quad
              \underline{a} \cap_\tau (\underline{a} \cup_\tau \underline{b})
              = \underline{a}.
          \]

    \item \textbf{Distributivity:}
          \[
              \underline{a} \cap_\tau (\underline{b} \cup_\tau \underline{c})
              = (\underline{a} \cap_\tau \underline{b})
              \cup_\tau (\underline{a} \cap_\tau \underline{c}),
          \]
          \[
              \underline{a} \cup_\tau (\underline{b} \cap_\tau \underline{c})
              = (\underline{a} \cup_\tau \underline{b})
              \cap_\tau (\underline{a} \cup_\tau \underline{c}).
          \]
\end{enumerate}

All five properties follow from the definitions
$\cup_\tau = \mathrm{lcm}$ and $\cap_\tau = \mathrm{gcd}$,
combined with the formula:
\[
    \mathrm{lcm}(a, b) = \prod_{p} p^{\max(v_p(a), v_p(b))},
    \quad
    \mathrm{gcd}(a, b) = \prod_{p} p^{\min(v_p(a), v_p(b))},
\]
where the products run over all primes
(Fundamental Theorem of Arithmetic, Theorem~\ref{thm:fta-tau-idx}).

\emph{Commutativity and associativity}
follow from commutativity and associativity
of $\max$ and $\min$.

\emph{Idempotence:}
$\max(v_p(a), v_p(a)) = v_p(a)$
and $\min(v_p(a), v_p(a)) = v_p(a)$.

\emph{Absorption:}
For union absorption,
\begin{align*}
    \underline{a} \cup_\tau (\underline{a} \cap_\tau \underline{b})
    &= \prod_{p} p^{\max(v_p(a), \min(v_p(a), v_p(b)))}\\
    &= \prod_{p} p^{v_p(a)}
    = \underline{a},
\end{align*}
since $\max(x, \min(x, y)) = x$ for all $x, y \in \mathbb{R}$.
Intersection absorption is dual.

\emph{Distributivity:}
For the first distributive law,
\begin{align*}
    &\underline{a} \cap_\tau (\underline{b} \cup_\tau \underline{c})\\
    &= \prod_{p} p^{\min(v_p(a), \max(v_p(b), v_p(c)))}\\
    &= \prod_{p} p^{\max(\min(v_p(a), v_p(b)),
    \min(v_p(a), v_p(c)))}
    \quad \text{(by distributivity of $\min$ over $\max$)}\\
    &= (\underline{a} \cap_\tau \underline{b})
    \cup_\tau (\underline{a} \cap_\tau \underline{c}).
\end{align*}
The second distributive law is dual
(using distributivity of $\max$ over $\min$).