Examples of magmas with all their elements idempotents

Question

A magma is supposed to be closed under a binary operation. Are there examples of magmas with all their elements idempotents under the operation of the magma?

Answer 1

That is rather simple to find such magmas because there is only one restriction on the binary operation of a magma: the operation must be closed. If (M,+) is a magma and M is a set with four elements, $M=\{a,b,c,d\}$, then we can draw the table for this operation. The main diagonal must be filled so that the operation is idempotent. for the remaining positions in the table there is no restriction except that the values must be from $M$.

+|a b c d
---------
a|a . . .   
b|. b . .
c|. . c .
d|. . . d

so there are $4^{12}$ idempotent magmas with 4 elements.

Answer 2

The multiplicative monoid of any Boolean ring would be an example.

This can be as small as the field of two elements $\Bbb F_2$, or even as large as the product ring $\prod_{i=1}^\infty \Bbb F_2$

Answer 3

Semilattices, or equivalently commutative idempotent semigroups, are magmas with all their elements idempotent. As an example, take the tropical algebra $(\mathbb{R} \cup \{-\infty\}, \max, +)$.

Answer 4

Just consider the multiplicative structures (monoids) over $\mathbb{Z}_2$ and $\mathbb{Z}_3$: $$\begin{array}{c|cc} \cdot_2 & 0 & 1 \\ \hline 0 & \color{lime}{0} & 0 \\ 1 & 0 & \color{lime}{1} \end{array} \qquad \begin{array}{c|ccc} \cdot_3 & 0 & 1 & 2 \\ \hline 0 & \color{lime}{0} & 0 & 0 \\ 1 & 0 & \color{lime}{1} & 2 \\ 2 & 0 & 2 & \color{red}{1} \end{array}$$ In $(\mathbb{Z}_2,\cdot_2)$ both elements are idempotent while in $(\mathbb{Z}_3,\cdot_3)$ only $0$ and $1$ are idempotent but $2$ is not. This is why we call only the former idempotent magma.