Definition:Addition of Order Types

Definition
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\alpha := \map {\operatorname {ot} } {S_1, \preccurlyeq_1}$ and $\beta := \map {\operatorname {ot} } {S_2, \preccurlyeq_2}$ denote the order types of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ respectively.

Let $\alpha + \beta$ be defined as:
 * $\alpha + \beta:= \map {\operatorname {ot} } {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} }$

where $\oplus$ denotes the order sum operator.