If a property is self-dual and holds for a lattice, does it hold for its dual

Question

I'm sorry for the silly question, but it is important. If a statement or definition is equivalent to its dual then it is said to be self-dual.

Now, if a property is self-dual and holds for a lattice, does it hold for its dual?

I have in mind distributivity or modularity, for which the above claim holds. But, in general, may I infer the above claim?

Answer 1

The answer is yes.

Let $\Phi$ be a property in the language of lattices, that is, involving the operations $\wedge$ and $\vee$.
Being self-dual means the property $\Phi^{\partial}$ obtained from $\Phi$ replacing every occurrence of $\wedge$ by $\vee$ and every occurrence of $\vee$ by $\wedge$, is equivalent to $\Phi$, that is, it holds in a lattice iff $\Phi$ holds in that lattice.

Now take a lattice $\mathbf L$ and a self-dual property $\Phi$, with $\mathbf L \models \Phi$.
Then $\mathbf L^{\partial} \models \Phi^{\partial}$, since $\mathbf L^{\partial}$ is also obtained from $\mathbf L$ by exchanging $\wedge$ and $\vee$.
Therefore $\mathbf L^{\partial} \models \Phi$.