generalized_boolean_algebras

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=====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====

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


====Subclasses====
[[Boolean algebras]] 


====Superclasses====
[[Brouwerian algebras]] 

[[Wajsberg hoops]] 


====References====

[(Ln19xx>
)]

Trace: • post_algebras • semirings_with_identity • normal_valued_lattice-ordered_groups • pseudo_mv-algebras • left_neofield • stone_algebras • lie_algebras • cancellative_semigroups • bck-algebras • join-semidistributive_lattices