%
\label{def:set-operations}
For $\underline{a}, \underline{b} \in \tau\text{-Idx}$,
we define:
\begin{enumerate}
    \item The \textbf{$\tau$-union} of $\underline{a}$ and $\underline{b}$
          by:
          \[
              \boxed{%
              \underline{a} \cup_\tau \underline{b}
              := \mathrm{lcm}(\underline{a}, \underline{b}).}
          \]
          This is the \emph{smallest} object
          (under $\in_\tau$)
          that contains both $\underline{a}$ and $\underline{b}$
          as members.

    \item The \textbf{$\tau$-intersection} of $\underline{a}$ and $\underline{b}$
          by:
          \[
              \boxed{%
              \underline{a} \cap_\tau \underline{b}
              := \mathrm{gcd}(\underline{a}, \underline{b}).}
          \]
          This is the \emph{largest} object
          (under $\in_\tau$)
          that is a member of both $\underline{a}$ and $\underline{b}$.
\end{enumerate}