Category:Ordered Sums

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This category contains results about Ordered Sums.


Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be tosets.

Let:

  • the order type of $\left({S, \preceq_1}\right)$ be $\theta_1$
  • the order type of $\left({T, \preceq_2}\right)$ be $\theta_2$.


Let $S \cup T$ be the union of $S$ and $T$.

We define the ordering $\preceq$ on $S$ and $T$ as:

$\forall s \in S, t \in T: a \preceq b \iff \begin{cases} a \preceq_1 b & : a \in S \land b \in S \\ a \preceq_2 b & : \neg \left({a \in S \land b \in S}\right) \land \left({a \in T \land b \in T}\right) \\ & : a \in S, b \in T \end{cases}$

That is:

If $a$ and $b$ are both in $S$, they are ordered as they are in $S$.
If $a$ and $b$ are not both in $S$, but they are both in $T$, they are ordered as they are in $T$.
Otherwise, that is if $a$ and $b$ are in both sets, their ordering in $S$ takes priority over that in $T$.


The ordered set $\left({S \cup T, \preceq}\right)$ is called the ordered sum of $S$ and $T$, and is denoted $S + T$.

Pages in category "Ordered Sums"

The following 2 pages are in this category, out of 2 total.