Boolean groups

A Boolean group is a monoid M = (M, ·, e) such that every element has order 2: x·x = e.

Let M and N be Boolean groups. A morphism from M to N is a function h : M→N that is a homomorphism: h(x·y) = h(x)·h(y) and h(e) = e.

({0,1}, + ,0), the two-element group with addition-mod-2. This algebra generates the variety of Boolean groups.

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
Definable principal congruences
Equationally definable principal congruences	no
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 0
[Size 4]?: 1
[Size 5]?: 0
[Size 6]?: 0
[Size 7]?: 0
[Size 8]?: 1

[Trivial algebras]?