Lexicographic

Lexicographic is an adjective with two closely related uses. In lexicography, it relates to the practice and theory of compiling dictionaries and studying words. In mathematics and computer science, it describes a method of ordering objects, known as lexicographic or dictionary order.

Etymology and scope are informative here: the term derives from the Greek words for word (lexis) and to write (graphein). As an adjective, lexicographic often appears in discussions of dictionaries, word lists, and the organization of linguistic data, as well as in technical contexts where a dictionary-style ordering is relevant.

In lexicographic order, objects are compared from the first component onward. For two sequences, the first position where they differ determines the order, and if one sequence is a proper prefix of the other, the shorter one is considered smaller. For strings, this yields intuitive results such as "apple" < "banana" and "aa" < "aab." The exact outcome depends on the chosen alphabet and its customary order of characters, so locale and encoding can influence the ordering.

Applications of lexicographic ordering are widespread. It underpins dictionary sorting, string sorting in programming and databases, and formal definitions in combinatorics and topology (for example, lexicographic order on product sets). The phrases “lexicographically smallest” or “lexicographic order” are common in both theoretical discussions and practical algorithms, highlighting a universal, left-to-right comparison method that aligns with human notions of dictionary arrangement.