Count of Binary Operations with Identity

Theorem

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

The number $N$ of possible different binary operations which have an identity element that can be applied to $S$ is given by:

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

Proof

From Count of Binary Operations with Fixed Identity, there are $n^{\left({n-1}\right)^2}$ such binary operations for each individual element of $S$.

As Identity is Unique, if $x$ is the identity, no other element can also be an identity.

As there are $n$ different ways of choosing such an identity, there are $n \times n^{\left({n-1}\right)^2}$ different algebraic structures with an identity.

These are guaranteed not to overlap by the uniqueness of the identity.

Hence the result.

$\blacksquare$

Comment

The number grows rapidly with $n$:

$\begin{array} {c|cr} n & \left({n-1}\right)^2 + 1 & n^{\left({n-1}\right)^2 + 1}\\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 5 & 243 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$

This sequence is A090602 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $4.2$