Definition:Lexicographic Order/Tuples of Equal Length/Cartesian Space

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Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $n \in \N_{>0}$.

Let $S^n$ be the cartesian $n$th power of $S$:

$S^n = \underbrace {S \times S \times \cdots \times S}_{\text {$n$ times} }$

The lexicographic order on $S^n$ is the relation $\preccurlyeq_l$ defined on $S^n$ as:

$\tuple {x_1, x_2, \ldots, x_n} \preccurlyeq_l \tuple {y_1, y_2, \ldots, y_n}$ if and only if:
$\exists k: 1 \le k \le n: \paren {\forall j: 1 \le j < k: x_j = y_j} \land \paren {x_k \prec y_k}$
$\forall j: 1 \le j \le n: x_j = y_j$

That is, if and only if:

the elements of a pair of $n$-tuples are either all equal


they are all equal up to a certain point, and on the next one they are comparable and they are different.

Also known as

Lexicographic order can also be referred to as the more unwieldy lexicographical ordering.

Some sources refer to it as dictionary order.

Some sources classify the lexicographic order as a variety of order product.

Hence the term lexicographic product can occasionally be seen.

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

Some sources suggest Lex, but this has yet to filter through to general usage.

Also see

  • Results about Lexicographic Order can be found here.