Table of Contents
Generalized Boolean algebras
Abbreviation: GBA
Definition
A \emph{generalized Boolean algebra} is a Brouwerian algebras $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that
$x\vee y=(x\rightarrow y)\rightarrow y$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be generalized Boolean algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | decidable |
| Locally finite | yes |
| Residual size | $2$ |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes, $n=2$ |
| Congruence regular | yes |
| Congruence e-regular | yes, $e=1$ |
| Congruence uniform | yes |
| Congruence extension property | yes |
| Definable principal congruences | yes |
| Equationally def. pr. cong. | yes |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &0
f(4)= &1
f(5)= &0
f(6)= &0
\end{array}$
