Abbreviation: BGrp

A \emph{Boolean group} is a monoid $\mathbf{M}=\langle M, \cdot, e\rangle$ such that

every element has order $2$: $x\cdot x=e$.

Let $\mathbf{M}$ and $\mathbf{N}$ be Boolean groups. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$

Example 1: $\langle \{0,1\},+ ,0\rangle$, the two-element group with addition-mod-2. This algebra generates the variety of Boolean groups.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &0 f(4)= &1 f(5)= &0 f(6)= &0 f(7)= &0 f(8)= &1 \end{array}$

Trivial algebras

Abelian groups