Abbreviation: LA$_n$

A \emph{Lukasiewicz algebra of order $n$} is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\neg,\sigma_0,\ldots,\sigma_{n-1}\rangle $ such that

$\langle A,\vee ,0,\wedge ,1, \neg\rangle $ is a De Morgan algebras

1. $\sigma_i$ is a lattice homomorphism: $\sigma_i(x\vee y)=\sigma_i(x)\vee\sigma_i(y) \mbox{and} \sigma_i(x\wedge y)=\sigma_i(x)\wedge\sigma_i(y)$

2. $\sigma_i(x) \vee \neg(\sigma_i(x)) = 1$, $\sigma_i(x) \wedge \neg(\sigma_i(x)) = 0$

3. $\sigma_i(\sigma_j(x)) = \sigma_j(x)$ for $1 \le j \le n-1$

4. $\sigma_i(\neg x) = \neg(\sigma_{n-i}(x))$

5. $\sigma_i(x) \wedge \sigma_j(x) = \sigma_i(x)$ for $i \le j \le n - 1$

6. $x \vee \sigma_{n-1}(x) = \sigma_{n-1}(x)$, $x \wedge \sigma_1(x) = \sigma_1(x)$

7. $y \wedge (x \vee \neg(\sigma_i(x)) \vee \sigma_{i+1}(y)) = y$ for $i \ne n - 1$

Let $\mathbf{A}$ and $\mathbf{B}$ be Lukasiewicz algebras of order $n$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(\neg x)=\neg h(x)$, $h(\sigma_i(x))=\sigma_i(h(x))$ for $i=0,\ldots,n-1$

Example 1:

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

Boolean algebras

De Morgan algebras