Count of Binary Operations Without Identity

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Theorem

Let $S$ be a set whose cardinality is $n$.

The number $N$ of possible different binary operations which do not have an identity element that can be applied to $S$ is given by:

$N = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$


Proof

From Count of Binary Operations on Set, the total number of operations is $n^{\left({n^2}\right)}$.

From Count of Binary Operations with Identity, the total number of operations with an identity is $n^{\left({n-1}\right)^2 + 1}$.

So the total number of operations without an identity is:

$n^{\left({n^2}\right)} - n^{\left({n-1}\right)^2 + 1} = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$


Hence the result.

$\blacksquare$


Comment

The number grows rapidly with $n$:

$\begin{array} {c||r|r|r} n & n^{\left({n^2}\right)} & n^{\left({n-1}\right)^2 + 1} & n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)\\ \hline 1 & 1 & 1 & 0 \\ 2 & 16 & 4 & 12 \\ 3 & 19 \ 683 & 243 & 19 \ 440 \\ 4 & 4 \ 294 \ 967 \ 296 & 1 \ 048 \ 576 & 4 \ 293 \ 918 \ 720\\ \end{array}$

This sequence is A175882 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Sources