Boolean groups

Abbreviation: BGrp

Definition

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

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

Morphisms

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:

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

Examples

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

Properties

Classtype variety decidable in polynomial time decidable decidable yes 2 no yes yes, $n=2$ yes yes yes no

Finite members

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