Ockham algebras

Abbreviation: OckA

Definition

An \emph{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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