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