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 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.

Sources

• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.4$