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hilbert_algebras [2010/07/29 15:46]
127.0.0.1 external edit
hilbert_algebras [2016/09/02 09:28] (current)
jipsen
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$x\to(y\to x)=1$ $x\to(y\to x)=1$
-$(x\to(y\to z))\to((x\to y)\to(x\to y))=1$+$(x\to(y\to z))\to((x\to y)\to(x\to z))=1$
$x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$ $x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$
- 
-Remark: This is a template. 
-If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page. 
- 
-It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes. 
==Morphisms== ==Morphisms==
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====Definition==== ====Definition====
-An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +A \emph{Hilbert algebra} is a structure $\mathbf{A}=\langle A,\to,1\rangle$ of type $\langle 2, 1\rangle$ such that
-...\rangle$ such that+
-$...$ is ...:  $axiom+$x\to x=1$ 
-   + 
-$...$ is ...:  $axiom$+$1\to x=x
 + 
 +$x\to(y\to z)=(x\to y)\to(x\to z)$ 
 + 
 +$(x\to y)\to((y\to x)\to x)=(y\to x)\to((x\to y)\to y)$
====Examples==== ====Examples====
-Example 1: +Example 1: Given any poset with top element 1, $\langle A,\le, 1\rangle$, define $a\to b=\begin{cases}1&\text{ if $a\le b$}\\ b&\text{ otherwise.}\end{cases}$ Then $\langle A,\to,1\rangle$ is a Hilbert algebra.
====Basic results==== ====Basic results====
 +Hilbert algebras are the algebraic models of the implicational fragment of [[wp>intuitionistic logic]], i.e., they are $(\to,1)$-subreducts of [[Heyting algebras]].
 +
 +The variety of Hilbert algebras is not generated as a quasivariety by any of its finite members [(CelaniCabrer2005)].
====Properties==== ====Properties====
-Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. 
^[[Classtype]]                        |variety [(Diego1966)]  | ^[[Classtype]]                        |variety [(Diego1966)]  |
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====Subclasses==== ====Subclasses====
-  [[...]] subvariety+[[...]] subvariety
-  [[...]] expansion+[[...]] expansion
====Superclasses==== ====Superclasses====
-  [[...]] supervariety+[[...]] supervariety
-  [[...]] subreduct+[[...]] subreduct
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[(Diego1966> [(Diego1966>
-A. Diego, \emph{Sur les algbres de Hilbert}, Collection de Logique Math\'ematique, S\'er. A, 1966, 1--55 [[MRreview]] +A. Diego, \emph{Sur les algébres de Hilbert}, Collection de Logique Math\'ematique, S\'er. A, 1966, 1--55 
 +)] 
 + 
 +[(CelaniCabrer2005> 
 +S. Celani and L. Cabrer: Duality for finite Hilbert algebras. Discrete Math. 305 (2005), no. 1-3, 74-–99.
)] )]