Generalization of the alphabetical order of dictionaries to sequences of elements of an ordered set
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For similarly named ordering systems outside mathematics, see alphabetical order, natural sort order, lexicographic preferences, and collation.
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.
and 25 Related for: Lexicographic order information
mathematics, the lexicographic or lexicographicalorder (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the...
forth ending with x1. The differences between graded lexicographic and graded reverse lexicographic orders are subtle, since they in fact coincide for 1...
methods of collation. In mathematics, a lexicographicalorder is the generalization of the alphabetical order to other data types, such as sequences of...
printed and electronic dictionaries One important goal of lexicography is to keep the lexicographic information costs incurred by dictionary users as low...
order. Each of these orders extends the next in the sense that if we have x ≤ y in the product order, this relation also holds in the lexicographic order...
instance. Thus the lexicographic successor of the initial state is permuted: [1, 2, 4, 3]. Following this algorithm, the next lexicographic permutation will...
Dictionary order may refer to: Alphabetical order § Treatment of multiword strings Other collation systems used to order words in dictionaries Lexicographic order...
In economics, lexicographic preferences or lexicographicorderings describe comparative preferences where an agent prefers any amount of one good (X) to...
total) order. As a classic example, the lexicographicorder of totally ordered sets is a linear extension of their product order. Every partial order can...
numbers, which are countable and can be placed in order first by sum of entries, and then by lexicographicorder within a given sum (only finitely many sequences...
graphs are order relations, then the edge relation of their lexicographic product is the corresponding lexicographicorder. The lexicographic product was...
has lexicographic preferences, ranking the possible solutions according to a lexicographicorder of their objective function values. Lexicographic optimization...
0)} are incomparable in the product order of the ordering 0 < 1 {\displaystyle 0<1} with itself. The lexicographic combination of two total orders is a...
An alternative construction involves concatenating together, in lexicographicorder, all the Lyndon words whose length divides n. An inverse Burrows–Wheeler...
In general topology, the lexicographicordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square...
or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus,...
leximin-equivalent (but both are leximin-larger than (2,4,2)). In the lexicographicorder, the first comparison is between x1 and y1, regardless of whether...
are sorted into lexicographicalorder. Shortlex ordering is also called radix, length-lexicographic, military, or genealogical ordering. In the context...
of a string possessing the lowest lexicographicalorder of all such rotations. For example, the lexicographically minimal rotation of "bbaaccaadd" would...
numbers correspond to distinct k-combinations, and produce them in lexicographicorder. The numbers less than ( n k ) {\displaystyle {\tbinom {n}{k}}} correspond...
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing...
: 73 Lexicographic sorting of a set of string keys can be implemented by building a trie for the given keys and traversing the tree in pre-order fashion;...
based algorithms for lexicographic breadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of...
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism...
a Lyndon word is a nonempty string that is strictly smaller in lexicographicorder than all of its rotations. Lyndon words are named after mathematician...
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