## Monday, April 21, 2014

### Lattice Rationals, 6 of 10

LCM and compensator: review of definitions and laws

lcm( product(pi^ai),  product(pi^bi)  )   =    product(pi^ max(ai, bi)  )
gcf( product(pi^ai),  product(pi^bi)  )   =    product(pi^ min(ai, bi)  )
( product(pi^ai) ;  product(pi^bi)  )   =    product(pi^ max(ai - bi , 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:
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