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generalized_boolean_algebras [2010/07/29 15:46] (current)
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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>
 +)]