I started thinking about the number of associative (binary) operations on a set with $n$ elements today. Looking online I found this paper which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association. So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association. For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small? I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does
$$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$
Also, is $F : n \to R_{n}$ a monotonically decreasing function?
Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different. Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it. Violations of the associative law, on the other hand, are usually considered by specialists only." For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say. But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them.
X
, when it is clear fromyour personal considerations
that they should do/sayY
instead?" (with a sixth possibly falling in the same category). One other was "X
saysthis
. Is that really true?" And one was "I did this clever thing! Does anyone know if I've been scooped?" You don't seem to be asking questions, you seem to be searching for either reassurance or praise. $\endgroup$