Lexicographic order no relation computers
Numeral systems and dates
The lexicographical order is used not only in dictionaries, but also commonly for numbers and dates.
One of the drawbacks of the Roman numeral system is that it is not always immediately obvious which of two numbers is the smaller. On the other hand, with the positional notation of the Hindu–Arabic numeral system, comparing numbers is easy, because the natural order on nonnegative integers is the same as the variant shortlex of the lexicographic order. In fact, with positional notation, a nonnegative integer is represented by a sequence of numerical digits, and an integer is larger than another one if either it has more digits (ignoring leading zeroes) or the number of digits is the same and the first (most significant) digit which differs is larger.
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. The padding 'blank' in this context is a trailing "0" digit.
When negative numbers are also considered, one has to reverse the order for comparing negative numbers. This is not usually a problem for humans, but it may be for computers (testing the sign takes some time). This is one of the reasons for adopting two's complement representation for representing signed integers in computers.
Another example of a non-dictionary use of lexicographical ordering appears in the ISO 8601 standard for dates, which expresses a date as YYYY-MM-DD. This formatting scheme has the advantage that the lexicographical order on sequences of characters that represent dates coincides with the chronological order: an earlier CE date is smaller in the lexicographical order than a later date up to year 9999. This date ordering makes computerized sorting of dates easier by avoiding the need for a separate sorting algorithm.Finite subsets
In combinatorics, one has often to enumerate, and therefore to order the finite subsets of a given set 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
   For this, one usually chooses an order on 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
   Then, sorting a subset of 
  
    
      
        S
      
    
    {\displaystyle S}
   is equivalent to convert it into an increasing sequence. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the lexicographical order.
In this context, one generally prefer to sort first the subsets by cardinality, such as in the shortlex order. Therefore, in the following, we will consider only orders on subsets of fixed cardinal.
For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of 
  
    
      
        S
        =
        {
        1
        ,
        2
        ,
        3
        ,
        4
        ,
        5
        ,
        6
        }
      
    
    {\displaystyle S=\{1,2,3,4,5,6\}}
   is
no relation