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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>
+)]