lexicographically

Lexicographically is an adverb describing the method of comparison or sorting by lexicographic order, also known as dictionary order. This order is defined on finite sequences (such as strings of letters or numbers, or tuples) drawn from a totally ordered alphabet.

For two sequences x and y, x is considered smaller than y if at the first position where they differ, the element of x is less than that of y; if all elements are equal up to the length of the shorter sequence, the shorter sequence is smaller. Examples: "apple" < "apply" because e < y at the last position; "cat" < "catalog" because "cat" is a prefix and shorter.

In mathematics and computer science, lexicographic order underpins dictionaries, string sorting, and tuple ordering. It is a total order, meaning every pair of sequences is comparable, and it is transitive and antisymmetric, providing a consistent and reproducible arrangement of items.

The term derives from lexicography, the study or practice of compiling dictionaries, itself from the Greek words lexis (word) and gráphein (to write). The adjective is lexicographic and the adverb is lexicographically.

Applications of lexicographic ordering appear in data processing, search algorithms, programming languages, and information retrieval, wherever stable, deterministic ordering of strings, lists, or sequences is required. It is distinct from but related to general alphabetical order used for human-friendly indexing.