Definition:Lexicographic Order/General Definition

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Definition

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

For $n \in \N: n > 0$, we define $T_n$ as the set of all ordered $n$-tuples:

$\left({x_1, x_2, \ldots, x_n}\right)$

of elements $x_j \in S$.

Let $\displaystyle T = \bigcup_{n \mathop \ge 1} T_n$.

The lexicographic order on $T$ is the relation $\preccurlyeq$ defined on $T$ as:

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


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

or:

all elements are equal up to the length of the shorter one.


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