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## Gödel algebras

Abbreviation: **GödA**

### Definition

A ** 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 ** linear Heyting algebras** since subdirectly irreducible G\”odel algebras are linearly ordered Heyting algebras.

### Definition

A ** 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 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

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ f(8)= &3\\ f(9)= &1\\ f(10)= &2\\ \end{array}$

### Subclasses

### Superclasses

### References

Trace: » goedel_algebras