## Ockham algebras

Abbreviation: OckA

### Definition

An Ockham algebra is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,'\rangle$ such that

$\langle A,\vee ,0,\wedge ,1\rangle$ is a bounded distributive lattice

$'$ is a dual endomorphism: $(x\wedge y)' =x'\vee y'$, $(x\vee y)' =x'\wedge y'$, $0'=1$, $1'=0$

##### Morphisms

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

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

Example 1:

### Properties

Classtype Variety Yes Yes

### Finite members

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

### References

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