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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.
The lexicographic product of graphs. In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the cartesian product V(G) × V(H); and; any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is ...
On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes impractical many computations that are relatively easy with graded reverse lexicographic order (grevlex), or, when elimination is needed, the elimination order (lexdeg) which restricts to grevlex on each block of variables.
The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. [3] The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
Order, an academic journal on order theory; Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order; Glossary of order theory; Lexicographical order, an ordering method on sequences analogous to alphabetical order on words; List of order topics, list of order theory topics
In mathematics, especially order theory, a partial order on a set is an arrangement such that, ... The lexicographical order on the cartesian product ...
Graded reverse lexicographic order (grevlex, or degrevlex for degree reverse lexicographic order) compares the total degree first, then uses a lexicographic order as tie-breaker, but it reverses the outcome of the lexicographic comparison so that lexicographically larger monomials of the same degree are considered to be degrevlex smaller.
In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length are sorted into lexicographical order. [1]
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