=====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====

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

====Subclasses====
[[Boolean algebras]] 

====Superclasses====
[[Heyting algebras]] 


====References====

[(Ln19xx>
)]