$ The quasiGroup existence problem (CSP lib problem 3)
$
$ An m order quasigroup  is an mxm multiplication table of integers 1..m, 
$ where each element occurrs exactly once in each row and column and certain 
$ multiplication axioms hold (in this case, we want axiom 3 to hold). 
$
$ bby Ian Miguel

ESSENCE' 1.0
given n : int(1..1000)
letting nDomain be domain int(0..n-1)

find quasiGroup : matrix indexed by [nDomain, nDomain] of nDomain
find qgDiagonal : matrix indexed by [nDomain] of nDomain

such that 
     $ accessor for diagonal
     forall i : nDomain .
         qgDiagonal[i] = quasiGroup[i,i] ,

     $ All rows have to be different
    forall row : nDomain . 
          allDifferent(quasiGroup[row,..]) ,

     $ All columns have to be different	       
     forall col : nDomain . 
          allDifferent(quasiGroup[..,col]),

    $  (j*i)*(i*j) = i
     forall i : nDomain .
         forall j : nDomain .	
	        quasiGroup[quasiGroup[i,j],quasiGroup[j,i]] = i $,

     $ Idempotency
     $forall i : nDomain . 
     $     (quasiGroup[i,i] = i), 

     $ Implied (from Colton,Miguel 01)
     $ All-diff diagonal
 $    allDifferent(qgDiagonal) $,

     $ anti-Abelian
    $ forall i : nDomain .
    $   forall j : nDomain .
    $     (i != j) =>
    $     (quasiGroup[i,j] != quasiGroup[j,i]),

     $ if (i*i)=j then (j*j) = i
     $forall i : nDomain .
     $  forall j : nDomain .
     $    (quasiGroup[i,i]=j) => (quasiGroup[j,j]=i),


     $ Symmetry-breaking constraints	
     $forall i : nDomain .
     $      quasiGroup[i,n-1] + 2 >= i