Definition:Ordered Sum/Caution
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Definition
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.