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 \and b \in S \\ a \preceq_2 b & : \neg \left({a \in S \and b \in S}\right) \and \left({a \in T \and 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 they are in different sets, the one that is in $$S$$ comes first.

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 $$\theta_1 + \theta_2$$.

The Ordered Sum of Tosets is a Totally Ordered Set.

General Definition
We can define the ordered sum of any finite number of tosets as follows.

Let $$S_1, S_2, \ldots, S_n$$ all be tosets.

Then we define $$T_n$$ as the ordered sum of $$S_1, S_2, \ldots, S_n$$ as:


 * $$\forall n \in \N^*: T_n = \begin{cases}

S_1 & : n = 1 \\ T_{n-1} + S_n & : n > 1 \end{cases}$$

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 Note
The ordered sum is defined only for totally ordered sets.