Lattice Rationals, 6 of 10

LCM and compensator: review of definitions and laws

                              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)  )

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

                              lcm(0,a) =             0

                              gcf(0,a)                 =             a

                              (0;a)                      =             0             if   a does not equal zero.

                              (a;0)                      =             (0;0)       =             1

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

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

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

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

                              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)

               From these definitions, you can derive these laws:

               Lcm and gcf are commutative and associative.

               Lcm and gcf double-distribute over each other.

               Absolute value distribution:

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

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

               Triple distribution:

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

               Alternation:        

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

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

               Duality:              

lcm(a,b) * gcf(a,b)     =      a*b

               Venn Laws:

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

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

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

               Consider a Venn diagram; two overlapping disks, representing  a and b. The union of the disks corresponds to lcm, the intersection of the disks corresponds to gcf, and the two moon-shaped regions correspond to the compensators.

                              (b;b)   =   1  if b is at least zero;       (b;b)   =   -1   if  b<0

                              (ab;ac)   =   (b;c)  if a is at least zero;       (ab;ac)   =   - (b;c)   if  a<0

                              (1;c)     =     1

                              (c;1)     =     c

(0;c)     =     0     if c is not zero;     (0;0)   =   1

                              (c;0)   =  1   if c>0  ;    (c;0) = 1    if c=0    ;     (c;0)  = -1  if c<0