Definition:Ordered Sum

Definition
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 $S + T$. The order type of $S + T$ is denoted $\theta_1 + \theta_2$.

General Definition
The ordered sum of any finite number of tosets is defined as follows:

Informal interpretation
We can consider the ordered set $\left({S \cup T, \preceq}\right)$ as:


 * First the whole of $S$, ordered by $\preceq_1$
 * After that, the set $T \setminus S$, ordered by $\preceq_2$, where $T \setminus S$ denotes set difference.

Caution
Note the way this definition 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.

Also see

 * Ordered Sum of Tosets is Totally Ordered Set