# Definition:Antilexicographic Order/Family

## Definition

Let $\struct {I, \preceq}$ be an ordered set such that its dual $\struct {I, \succeq}$ is well-ordered.

For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an ordered set.

Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.

Then the **antilexicographic order** on $D$ is defined as:

- $\ds \struct {D, \preccurlyeq_D} := {\bigotimes_{i \mathop \in I} }^a \struct {S_i, \preccurlyeq_i}$

where $\preccurlyeq_D$ is defined as:

- $\forall u, v \in D: u \preccurlyeq_D v \iff \begin {cases} u = v \\ \exists i \in I: \paren {\forall j > i: \map u j = \map v j \text { and } \map u i \preccurlyeq_i \map v i} \end {cases}$

## Also known as

**Antilexicographic order** can also be referred to as **colexicographic order**.

Some sources classify the **antilexicographic order** as a variety of order product.

Hence the term **antilexicographic product** can occasionally be seen.

Some sources which focus more directly on real analysis refer to this merely as the **ordered product**, or **order product**.

This is because the other types of **order product** as documented on $\mathsf{Pr} \infty \mathsf{fWiki}$ are not of such importance in that context.

Such sources may even define the **order product** on a totally ordered set, ignoring its definition on the general ordered set.

The mathematical world is crying out for a less unwieldy term to use.

Some sources suggest **AntiLex** or **CoLex**, but this has yet to filter through to general usage.

## Also see

- Results about
**the antilexicographic order**can be found here.

## Sources

- 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: Exercise $37$