Lattice Rationals, 2 of 10

LCM and compensator for positive numbers

               We can define compensator, lcm and gcf from prime factorizations:

               lcm( product(p_i^a_i),  product(p_i^b_i)  )   =    product(p_i^ max(a_i, b_i)  )

               gcf( product(p_i^a_i),  product(p_i^b_i)  )   =    product(p_i^ min(a_i, b_i)  )

               ( product(p_i^a_i) ;  product(p_i^b_i)  )   =    product(p_i^ max(a_i - b_i , 0)  )

               Compensator, lcm and gcf follow these rules for positive numbers:

               lcm(b,d)      =      (d;b)*gcf(b,d)*(b;d)      =      (d;b)*b      =     (b;d)*d

               d   =   (d;b)*gcf(b,d)                          :              b   =   (b;d)*gcf(b,d)

               gcf(b,b)   =   b          ;          lcm(b,b)   =  b                     ;              (b;b)  =  1

               gcf(b,bc)   =   b        ;          lcm(b,bc)   =  bc                 ;              (b;bc)  =  1           ;              (bc;b)  =  c

               gcf(1,c)   =   1          ;          lcm(1,c)   =  c                      ;              (1;c)  =  1              ;              (c;1)  =  c

               Distribution laws:

                              gcf(a, lcm(b,c))   =  lcm(gcf(a,b),  gcf(ac))   

lcm(a, gcf(b,c))   =  gcf(lcm(a,b),  lcm(ac))                 

                              a*lcm(b,c)   =  lcm(a*b,  a*c)                                      

                              a*gcf(b,c)   =  gcf(a*b,  a*c)                         

These last two laws are because addition distributes over minimum and maximum, and lcm and gcf are defined by minima and maxima of exponents. For instance:

(2^3 * 3^5) * lcm(2^2 * 3^1,  2^1 * 3^4)  

=   (2^3 * 3^5) * 2^max(2,1)  * 3^max(1,4))

 =   2 ^ (  3 + max(2,1) )  *  3 ^ ( 5 + max(1,4) )

=   2 ^  max(3+2, 3+1) )  *  3 ^ max(5+1, 5+4)

=  lcm ( 2^(3+2) * 3^(5+1)  ,   2^(3+1) * 3^(5+4)  )

=  lcm (( 2^3 * 3^5)*(2^2  * 3^1)  ,   ( 2^3 * 3^5)*(2^1  * 3^4)   )

Also, the distribution of multiplication over lcm implies Cancellation:

(a*b ; a*d)    =   (b ; d)

Commutativity and associativity:

lcm(a,b) = lcm(b,a)            ;              lcm(a,lcm(b,c))    =    lcm(lcm(a,b),c))

gcf(a,b) = gcf(b,a)              ;              gcf(a,gcf(b,c))    =    gcf(gcf(a,b),c))

               These laws, plus the compensator definition of addition, yield “Semicancellation”:

                              (a*c)/(b*c)                         =                            a/b    +    0/(bc)

               I also call this law “casting out zeroids”, where a ‘zeroid’ is a ratio with numerator zero.

               This in turn implies “Compensated Reduction”:

                              a / b                                     =                            (a;b) / (b;a)   +  0/b

               Since lcm(b,b) = b   and  (b;b)=1, we recover the one-denominator rule:

                              (a/b) + (c/b)   =   (a*(b;b) + c*(b;b)) / lcm(b,b)   =   (a+c)/b

               But if b and d are relatively prime, then lcm(b,d)=b*d;   (b;d)=b;  and (d;b)=d; so we recover the two-denominators rule:

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