From: qscgz@aol.com (QSCGZ)
Subject: Re: Pseudogroup
Date: 18 Nov 1999 12:56:10 GMT
Newsgroups: sci.math
Keywords: smallest non-associative pseudogroup

 On 17 Nov 1999, Hauke Reddmann wrote:
 
 > Let's define a pseudogroup:
 > 
 > If in the multiplication table in every row and column
 > every element occurs exactly once, and if a neutral
 > element exists (place it in the first row/column for
 > convenience), and if this element occurs only in the
 > main diagonal or in mirror-pairs along it, the table
 > constitutes a pseudogroup. 
 > 
 > (All group axioms fulfilled except associative law, right?)
 > 
 > 1. Which is the smallest pseudogroup?
 > 1. Which is the smallest commutative pseudogroup?

For n<5 each pseudogroup is a group.
Pseudogroups over {1..n} for n=5:

  1 2 3 4 5
  2 1 4 5 3
  3 5 1 2 4
  4 3 5 1 2
  5 4 2 3 1

and it's transposed. Also 6 associative ones.
I just checked the 56 normalized latin squares of order 5.

For n=6 , there are 9408 normalized latin squares , 1808 of which
satisfy this 1-diagonal-condition , 80 of which are associative.

For n=7 , there are 16942080 normalized latin squares , which
I'm not going to test with this method now .

Statistically , I would assume that the ratio :
pseudogroups/(normalized latin squares) tends to R(n-1) , 
where R() satisfies the recursion 
i*R(i)=R(i-1)+R(i-2) with R(1)=R(2)=1.
Since the number of latin squares grows very fast , 
this additional factor is not very important for big n.
For e.g. n=15 this calculation gives about 10^88 reduced latin squares
and 10^83 pseudogroups over {1,2,..,15}.
Lots of them are isomorphic.



--qscgz