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