mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and particularly in the theory of

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

s, shortlex is a

total ordering In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...

for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

(length) with the shortest sequences first, and sequences of the same length are sorted into

lexicographical order 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 ...

. Shortlex ordering is also called radix, length-lexicographic, military, or genealogical ordering. In the context of strings on a totally ordered alphabet, the shortlex order is identical to the lexicographical order, except that shorter strings precede longer strings. For example, the shortlex order of the set of strings on the English alphabet (in its usual order) is 'ε, a, b, c, ..., z, aa, ab, ac, ..., zz, aaa, aab, aac, ..., zzz, ...'' where ε denotes the

empty string In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), characters such as letters, digits ...

. The strings in this ordering over a fixed finite alphabet can be placed into one-to-one

order-preserving In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...

correspondence with the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s, giving the

bijective numeration Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection (i.e. one-to-one correspondence) that exists in this case betw ...

system for representing numbers. The shortlex ordering is also important in the theory of

automatic group In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell whether a given word representation of a group element is ...

s..

References

Order theory {{algebra-stub

See also

References