Lattice Rationals, 5 of 10

LCM and compensator for negative numbers and ratios

               For simplicity’s sake, and to resemble the 2-denominator rule as much as possible, I propose that  lcm is an odd function, and gcf is an even function:

               lcm(-a,b) = lcm(a,-b)   =  - lcm(a,b)

               gcf(-a,b) = gcf(a,-b)   = gcf(a,b)

               so

               lcm(-a,-b) =  lcm(a,b)      ;     gcf(-a,-b)  =  gcf(a,b)

               and  as before:

lcm(a,b)*gcf(a,b)  =   a*b

               We no longer have double-distribution of multiplication over lcm:

                              -1 * lcm(2,3)  =  -6      but     lcm(-2,-3)  =  6

               But we still have triple-distribution:

                              a * lcm(b,c,d)      =    lcm(ab,ac,ad)

               and lcm and gcf still double-distribute over each other.

               We still have:

                              lcm(ab,ac)    =   |a|*lcm(b,c)

                              gcf(ab,ac)    =   |a|*gcf(b,c)

                              a * lcm(b,c)    =   sign(a)*lcm(ab,ac)

               We also have “Alternation”:        

                              lcm(a,a)        =    gcf(a,a)        =     |a|

                              lcm(a,|a|)    =    gcf(a,|a|)    =       a

               The compensator is odd in the first variable and even in the second:

                              (-a;b)    =    - (a;b)      for a not equal to zero.

                              (a;-b)    =      (a;b)

               All this, plus the Compensated Addition Rule, implies:

                              (a/-b)  +  (c/-d)      =     (-a/b)  +  (-c/d)

               In particular, consider 0/-1; call it the “alternator” @. Then:

                              (a/-b)   + @       =      (-a/b)

                              (a/b)   +  @       =      (-a/-b)

               Note that 0 and @ form a modulo-2 group under both addition and multiplication:

                              0+0      =    @+@    =    0*0      =    @*@    =    0  ;

                              0+@    =    @+0      =    0*@    =    @*0     =     @ 

               Multiplication by @ does not double-distribute over addition;

                              @ *(0+0)    =   @*0   =   @  ;     but   @*0 + @*0    =   @+@   =   0

               But it does triple-distribute:

                              @*(a+b+c)    =    @*a  +  @*b  +  @*c

               Define lcm, gcf and compensator of ratios as ratios:

                              Lcm(a/A,  b/B)     =   lcm(a,b) / lcm(A,B)

                              gcf(a/A,  b/B)     =   gcf(a,b) / gcf(A,B)

                              (a/A ;  b/B)     =   (a;b) / (A;B)

               We get many of the same rules as above; for instance lcm*gcf = product.