\begin{definition}[Definition:Ordered Sum/Caution]
Note the way the definition of an '''ordered sum''' has been worded.
Suppose $a, b \in S \cap T$.
Suppose:
:$a \preceq_1 b$ (through dint of $a, b \in S$
:$b \preceq_2 a$ (through dint of $a, b \in T$.
Then because $a, b \in S$, we have that $a \prec b$.
But, also because $a, b \in S$, we do not consider the fact that $a, b \in T$ and so the relation $b \preceq_2 a$ is ignored.
Category:Definitions/Ordered Sums
\end{definition}