Abbreviation: GBA

A \emph{generalized Boolean algebra} is a Brouwerian algebras $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that

$x\vee y=(x\rightarrow y)\rightarrow y$

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:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$

Example 1:

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

Boolean algebras

Brouwerian algebras

Wajsberg hoops