Lexicographic order
generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters
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 sequences 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 subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied.
A generalization defines an order on an n-ary Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
Definition
The words in a lexicon (the set of words used in some language) have a conventional ordering, used in dictionaries and encyclopedias, that depends on the underlying ordering of the alphabet of symbols used to build the words. The lexicographical order is one way of formalizing word order given the order of the underlying symbols.
The formal notion starts with a finite set A, often called the alphabet, which is totally ordered. That is, for any two symbols a and b in A that are not the same symbol, exactly one of a < b or b < a is true.
The words of A are the finite sequences of symbols from A, including words of length 1 containing a single symbol, words of length 2 with 2 symbols, and so on, even including the empty sequence
ε
{\displaystyle \varepsilon }
with no symbols at all.
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This entry incorporates text from “Lexicographic order” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.