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