Category:Ordered Sums

This category contains results about Ordered Sums.
Definitions specific to this category can be found in Definitions/Ordered Sums.

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let:

the order type of $\struct {S, \preceq_1}$ be $\theta_1$

the order type of $\struct {T, \preceq_2}$ 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 \paren {a \in S \land b \in S} \land \paren {a \in T \land b \in T} \\ & : 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 $\struct {S \cup T, \preceq}$ is called the ordered sum of $S$ and $T$, and is denoted $S + T$.