Editing Lexicographic order (section)

{{short description|Generalization of the alphabetical order of dictionaries to sequences of elements of an ordered set}}
{{more citations needed|date=January 2022}}
{{Hatnote|For similarly named ordering systems outside mathematics, see [[alphabetical order]], [[natural sort order]], [[lexicographic preferences]], and [[collation]].}}

In [[mathematics]], the '''lexicographic''' or '''lexicographical order''' (also known as '''lexical order''', or '''dictionary order''') is a generalization of the [[alphabetical order]] of the [[dictionaries]] to [[sequence]]s of ordered symbols or, more generally, of elements of a [[totally ordered set]].

There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements.

Another variant, widely used in [[combinatorics]], orders [[subset]]s of a given [[finite set]] by assigning a total order to the finite set, and converting subsets into [[Sequence#Increasing_and_decreasing|increasing sequences]], to which the lexicographical order is applied. 

A generalization defines an order on an ''n''-ary [[Cartesian product]] of [[partially ordered set]]s; this order is a total order if and only if all factors of the Cartesian product are totally ordered.