The Smarandache Groupoid, Ring, and Field

                  
                         Dwiraj Talukdar
                         Head Department of Mathematics
                         Nalbari College
                         Nalbari, Assam, India



Definition 1:
  
  Let the set Zp = {0, 1, 2, ..., p-1}.  The elements of Zp can be written 
uniquely as m-adic numbers.

Let r = ( a   a   ... a a  )   and  s = ( b   b   ... b b  )  be two
           n-1 n-2     1 0  m              n-1 n-2     1 0  m   

elements from Zp.  Then one introduces the operation:

  r /-\ s = (|a   - b   ||a   - b   | ... |a - b ||a - b |) 
               n-1   n-1   n-2   n-1        1   1   0   0  m
  
where |x-y| is the absolute value of x-y.
The set ( Zp, /-\ ) is known as Smarandache Groupoid.

  The complement of r, C(r) = r /-\ Sup(Zp), where Sup(Zp) is the maximal
element of Zp.

We say that rRs, iff r /-\ C(r) = s /-\ C(s).  R is a relation of 
equivalence which partitions Zp into equivalence classes.  The below
equivalence class is defined as D-form:

        D       = { r in Zp: r /-\ C(r) = Sup(Zp) }.
         Sup(Zp)


Definitions 2-3:

  Let's introduce two more operations on Zp:

  r /+\ s = (\a   + b   \\a   + b   \ ... \a + b \\a + b \) 
               n-1   n-1   n-2   n-1        1   1   0   0  m
  
where \x+y\ is the remainder of x+y modulo m.

  r /*\ s = (\a   * b   \\a   * b   \ ... \a * b \\a + b \) 
               n-1   n-1   n-2   n-1        1   1   0   0  m
  
where \x*y\ is the remainder of x*y modulo m.

Then the set ( Zp, /+\, /*\ ) is known as the Smarandache Ring.
If m is prime, then the set ( Zp, /+\, /*\ ) is known 
as the Smarandache Field.                                                         

Questions:
  
  1) Study the Smarandache Ring, and the Smarandache Field.

  2) If one introduces other operations on Zp:

  r /^\ s = (\a   ^ b   \\a   ^ b   \ ... \a ^ b \\a ^ b \) 
               n-1   n-1   n-2   n-1        1   1   0   0  m
  
where \x^y\ is the remainder of x^y (x to the power y) modulo m.
Study ( Zp, /^\ ).

   3) And in a similar way for any introduced digit operation on Zp:

  r /F\ s = (\a   F b   \\a   F b   \ ... \a F b \\a F b \) 
               n-1   n-1   n-2   n-1        1   1   0   0  m
  
where \xFy\ is the the remainder of xFy (i.e. F(x,y), for F a given
function, F: N -> N) modulo m.