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In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the way words are alphabetically ordered based on the alphabetical order of their component letters. This generalization consists primarily in defining a total order on the sequences (often called strings in computer science) of elements of a finite totally ordered set, often called an alphabet. There are several variants and generalizations of the lexicographical ordering. One 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. Another generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if the factors of the Cartesian product are totally ordered.
References
- Egbert Harzheim (2006). Ordered Sets. Springer. pp. 88–89. ISBN 978-0-387-24222-4.
- Franz Baader; Tobias Nipkow (1999). Term Rewriting and All That. Cambridge University Press. pp. 18–19. ISBN 978-0-521-77920-3.
- Calude, Cristian (1994). Information and randomness. An algorithmic perspective. EATCS Monographs on Theoretical Computer Science. Springer-Verlag. p. 1. ISBN 3-540-57456-5. https://archive.org/details/informationrando00calu_163.
- Robbiano, L. (1985). Term orderings on the polynomial ring. In European Conference on Computer Algebra (pp. 513-517). Springer Berlin Heidelberg.
- Weispfenning, Volker (May 1987), "Admissible Orders and Linear Forms", SIGSAM Bulletin (New York, NY, USA: ACM) 21 (2): 16–18, doi:10.1145/24554.24557 . https://dx.doi.org/10.1145%2F24554.24557
