Table of Contents
Gödel algebras
Abbreviation: GödA
Definition
A \emph{Gödel algebra} is a Heyting algebras $\mathbf{A}=\langle A,\vee,0,\wedge,1,\rightarrow\rangle$ such that
$(x\to y)\vee(y\to x)=1$
Remark: Gödel algebras are also called \emph{linear Heyting algebras} since subdirectly irreducible Gödel algebras are linearly ordered Heyting algebras.
Definition
A \emph{Gödel algebra} is a representable FLew-algebra $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \cdot, \rightarrow\rangle$ such that
$x\wedge y=x\cdot y$
Morphisms
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:
$h(x\vee y)=h(x)\vee h(y)$, $h(0)=0$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
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
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &1
f(6)= &2
f(7)= &1
f(8)= &3
f(9)= &1
f(10)= &2
\end{array}$
