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:
- $\forall u, v \in D: u \preccurlyeq_D v \iff \begin {cases} u = v \\
\exists i \in I: \paren {\forall j > i: \map u j = \map v j \text { and } \map u i \preccurlyeq_i \map v i} \end {cases}$
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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 $37$