These six procedures implement various order relations on strings, extended from the numeric order on character code points.

Each procedure takes two strings s1 and s2 as arguments, and returns one of the values , according to the following table.

The LexOrder(s1, s2) command implements simple lexicographic (dictionary) ordering of strings.

The RevLexOrder(s1, s2) command implements the reverse lexicographic order, which is equivalent to the lexicographic order on the reversed strings.

The ShortLexOrder(s1, s2) and ShortRevLexOrder(s1, s2) commands are similar, but apply the respective ordering relation only to strings of equal length. Strings are first compared according to their lengths, with shorter strings preceeding longer strings. The short variants are important because they are translation-invariant:  = ,  = ,  =  and  = ,

The left and right recursive path orders are also translation invariant, and are computed by the LeftRecursivePathOrder(s1, s2) and RightRecursivePathOrder(s1, s2) commands, respectively.

The left and right recursive path orders, manifest in the LeftRecursivePathOrder(s1, s2) and RightRecursivePathOrder(s1, s2) commands and are defined for two strings  and , as follows. Characters (strings with length equal to 1) compare according to the numeric order of the corresponding code points.

String  precedes string  in the left recursive path order if one among the following holds: (1)  and  precedes ; (2)  and  precedes ; or (3)  and  precedes .

String  precedes string  in the right recursive path order if one among the following holds: (1)  and  precedes ; (2)  and  precedes ; or (3)  and  precedes .