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G\"odel algebras
Abbreviation: G\”odA
Definition
A G\”odel algebra is a Heyting algebras $\mathbf{A}=\langle A,\vee,0,\wedge,1,\rightarrow\rangle$ such that
$(x\to y)\vee(y\to x)=1$
Remark: G\”odel algebras are also called linear Heyting algebras since subdirectly irreducible G\”odel algebras are linearly ordered Heyting algebras.
Definition
A G\”odel algebra is a representable FLew-algebra $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \cdot, \rightarrow\rangle$ such that
$x\wedge y=x\cdot y$
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 homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(0)=0$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
Examples
Example 1:
Basic results
Properties
Classtype | variety |
---|---|
Equational theory | decidable |
Quasiequational theory | decidable |
First-order theory | |
Locally finite | |
Residual size | countable |
Congruence distributive | yes |
Congruence modular | yes |
Congruence n-permutable | yes, $n=2$ |
Congruence e-regular | yes, $e=1$ |
Congruence uniform | |
Congruence extension property | yes |
Definable principal congruences | yes |
Equationally def. pr. cong. | yes |
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » goedel_algebras