أعمل حاليا على دعم المعادلات الرياضياتية العربية في ليبرأوفيس، النتائج مبشرة حتى الآن.
The number of groups with n elements goes like this, starting with n = 0:
0, 1, 1, 1, 2, 1, 2, 1, 5, ...
The number of semigroups with n elements goes like this:
1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292, ...
Here I'm counting isomorphic guys as the same.
Is there any sort of algebraic gadget where the number of them with n elements goes like this?
1, 1, 2, 1, 1, 1, 1, 1, 1, ...
No!
Not if by "algebraic gadget" we mean a thing defined to have finitely many operations which are required to obey finitely many equational laws.
(Of course one can build other operations from the finitely many given, and also derive more equations from those given.)
Apparently this follows from a result of László Lovász in 1967:
https://mathoverflow.net/q/454146/2893
The sequence I showed you has a(2²) < a(2). But he seems to have showed that if a(n) is the number of algebraic gadgets with n elements, we must have
a(n²) ≥ a(n)
because if A and B are two gadgets with A² ≅ B², then A ≅ B.
I say "apparently" and "seems" because the one paper by him on this stuff is hard for me to read. But this result sure seems nice! Does it become obvious if you look at it the right way? Like, with some category theory?
Substances for meditation enhancement are a 2×2, {low substance, high substance}×{low meditation, high meditation}.
High substance has problems: Insights don't stick/are vacuous, + dangerous
High meditation has opportunity costs: Learning rate and energy parameter are too low
Low substance and low meditation are just less effective modulo relaxation effects
But where is the pareto frontier?
I operate by Crocker's rules[1].