Definition:Multiplication of Order Types

Definition

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

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

Let $\alpha \cdot \beta$ be defined as:

$\alpha \cdot \beta:= \map \ot {\struct {S_1, \preccurlyeq_1} \otimes^a \struct {S_2, \preccurlyeq_2} }$

where $\otimes^a$ denotes the antilexicographic product operator.

The operation $\cdot$ is known as order type multiplication or multiplication of order types.

The expression $\alpha \cdot \beta$ is known as the order product of $\alpha$ and $\beta$.

Examples

Multiplication of Order Types/Examples

Sources

• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.4$: Ordered sums and products of ordered sets
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations