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 Gödel 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 Gödel algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(0)=0$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$

Example 1:

Properties

Classtype variety decidable decidable countable yes yes yes, $n=2$ yes, $e=1$ yes yes yes

Finite members

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