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:


 * $N = n^{\paren {n - 1}^2 + 1}$

Proof
From Count of Binary Operations with Fixed Identity, there are $n^{\paren {n - 1}^2}$ 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 $n \times n^{\paren {n - 1}^2}$ different magmas with an identity.

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

Hence the result.