There is a Wikipedia article about interior algebras. An interior algebra is a Boolean algebra with an additional unary operator, the interior operator, satisfying certain additional axioms. The axioms are dual to the Kuratowski closure axioms.
A Boolean algebra has a unary operator called complementation. Any nonzero element x of a (non-degenerate) Boolean algebra is distinct from (not equal to) its complement. (Proof: the meet of x with itself is x, while the meet of x with its complement is 0.)
One might think that in an interior algebra, we would have a similar proof that the interior of x is distinct from x, if only for elements satisfying certain conditions. But there appears to be no such proof.
Is there a proof I've overlooked? Is there a proof there's no proof?
And in general this seems odd; if anyone can explain why it's not that would be helpful.