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

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Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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

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

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


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

$\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ if and only if:
$\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec y_k}\right)$
or:
$\forall j: 1 \le j \le n: x_j = y_j$


That is, if and only if:

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

or:

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 known as the more unwieldy lexicographical ordering.


Also see

  • Results about Lexicographic Order can be found here.


Sources