This question by Moshe Newman:
How many different lattices are there on n points, that require exactly 3 elements to generate them? This sequence seems to start 0,0,1,0,4,3 (for n = 1 to 6) and seems not be be in the OEIS.
$\begingroup$
$\endgroup$
5
-
2$\begingroup$ It is a little confusing to say "on $n$ points" when what you mean is $n$-element lattices up to isomorphism. $\endgroup$– Richard StanleyCommented Feb 15, 2019 at 19:23
-
$\begingroup$ I can only guess what the question is. Do you mean to classify the pairs $(H, Z^n)$ consisting of a subgroup $H$ of standard $Z^n$ \st $Z^n/H$ is exactly three-generated (as an abelian group), up to GL($n, Z$) acting on $Z^n$? That is, the number of equivalence classes ... [Here $Z$ means the integers; I was too lazy to use bold $Z$, because MO doesn't allow macros to be defined.] $\endgroup$– David HandelmanCommented Feb 15, 2019 at 19:48
-
1$\begingroup$ @DavidHandelman this is about partially ordered sets that have meets and joins $\endgroup$– Bjørn Kjos-HanssenCommented Feb 15, 2019 at 20:30
-
1$\begingroup$ @BjørnKjos-Hanssen Ahh, that explains it. I am currently thinking about sublattices of $R^n$ (meaning discrete subgroups), hence my confusion. Whenever lattice is mentioned, it should be disambiguated. $\endgroup$– David HandelmanCommented Feb 15, 2019 at 21:45
-
$\begingroup$ Languages like French are more sensible: reseau vs. treillis. $\endgroup$– Richard StanleyCommented Feb 20, 2019 at 0:20
Add a comment
|