Count of Binary Operations with Identity

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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:

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


Sequence of Values of $N$

The sequence of $N$ for each $n$ begins:

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


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.

$\blacksquare$


Examples

Order $2$ Structure

The Cayley tables for the complete set of magmas of order $2$ which have an identity element are listed below.

The underlying set in all cases is $\set {a, b}$.

$\begin{array}{r|rr}
 & a & b \\

\hline a & a & a \\ b & a & b \\ \end{array}$

$\begin{array}{r|rr}
 & a & b \\

\hline a & a & b \\ b & b & a \\ \end{array} \qquad \begin{array}{r|rr} 

 & a & b \\

\hline a & a & b \\ b & b & b \\ \end{array}$

$\begin{array}{r|rr}
 & a & b \\

\hline a & b & a \\ b & a & b \\ \end{array}$


Sources

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