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goedel_algebras [2010/07/29 15:46]
127.0.0.1 external edit
goedel_algebras [2017/02/12 07:27] (current)
jipsen
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-=====G\"odel algebras=====+=====Gödel algebras=====
-Abbreviation: **G\"odA**+Abbreviation: **GödA**
====Definition==== ====Definition====
-A \emph{G\"odel algebra} is a [[Heyting algebras]] $\mathbf{A}=\langle A,\vee,0,\wedge,1,\rightarrow\rangle$ such that+A \emph{Gödel algebra} is a [[Heyting algebras]] $\mathbf{A}=\langle A,\vee,0,\wedge,1,\rightarrow\rangle$ such that
$(x\to y)\vee(y\to x)=1$ $(x\to y)\vee(y\to x)=1$
Remark: Remark:
-G\"odel algebras are also called \emph{linear Heyting algebras} since subdirectly irreducible G\"odel algebras are linearly ordered Heyting algebras.+Gödel algebras are also called \emph{linear Heyting algebras} since subdirectly irreducible Gödel algebras are linearly ordered Heyting algebras.
====Definition==== ====Definition====
-A \emph{G\"odel algebra} is a [[representable FLew-algebra]] $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \cdot, \rightarrow\rangle$ such that+A \emph{Gödel algebra} is a [[representable FLew-algebra]] $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \cdot, \rightarrow\rangle$ such that
$x\wedge y=x\cdot y$ $x\wedge y=x\cdot y$
==Morphisms== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be G\"odel algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a+Let $\mathbf{A}$ and $\mathbf{B}$ be Gödel algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism: homomorphism:
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f(2)= &1\\ f(2)= &1\\
f(3)= &1\\ f(3)= &1\\
-f(4)= &\\ +f(4)= &2\\ 
-f(5)= &\\ +f(5)= &1\\ 
-f(6)= &\\ +f(6)= &2\\ 
-f(7)= &\\ +f(7)= &1\\ 
-f(8)= &\\ +f(8)= &3\\ 
-f(9)= &\\ +f(9)= &1\\ 
-f(10)= &\\+f(10)= &2\\
\end{array}$ \end{array}$