The Cartesian product, " S�T ", of two well-ordered sets " S " and " T " can be well-ordered by a variant of lexicographical order that puts the least significant position first.
42.
Central to this approach is the view of the set of all executions of a statement within " n " loops as a set of integer points in an " n "-dimensional space, with the points being executed in lexicographical order.
43.
This well-order, sometimes called " shortlex " or " quasi-lexicographical order ", consists in considering first the lengths of the words ( if, then ), and, if the lengths are equal, using the lexicographical order.
44.
This well-order, sometimes called " shortlex " or " quasi-lexicographical order ", consists in considering first the lengths of the words ( if, then ), and, if the lengths are equal, using the lexicographical order.
45.
For real numbers written in decimal notation, a slightly different variant of the lexicographical order is used : the parts on the left of the decimal point are compared as before; if they are equal, the parts at the right of the decimal point are compared with the lexicographical order.
46.
For real numbers written in decimal notation, a slightly different variant of the lexicographical order is used : the parts on the left of the decimal point are compared as before; if they are equal, the parts at the right of the decimal point are compared with the lexicographical order.
47.
The "'closed long ray " "'L " is defined as the cartesian product of the half-open interval [ 0, 1 ), equipped with the order topology that arises from the lexicographical order on ? 1 & times; [ 0, 1 ).
48.
Note that for 2 d " " n " d " 9 the numerical order of the numbers 10 \ uparrow ^ m n is the lexicographical order with " m " as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line.
49.
On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes unpractical 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.
50.
There is a natural mapping between the integers 0, . . ., " n " ! " 1 ( or equivalently the numbers with " n " digits in factorial representation ) and permutations of " n " elements in lexicographical order, when the integers are expressed in factoradic form.
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