=====Boolean groups=====

Abbreviation: **BGrp**
====Definition====
A \emph{Boolean group} is a [[monoid]] $\mathbf{M}=\langle M, \cdot, e\rangle$ such that

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

==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.


====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable in polynomial time |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  |decidable |
^[[Locally finite]]  |yes |
^[[Residual size]]  |2 |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |

====Finite members====

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


====Subclasses====
[[Trivial algebras]] 

====Superclasses====
[[Abelian groups]] 


====References====

[(Ln19xx>
)]