Simple Ockham algebras (Q1917814)

It is proved that, for a finite simple Ockham algebra \((L;f)\) with \(n\) atoms, (1) if \(L\) has no fixed points then \(n\) is odd and \(L\simeq\mathbf{2}^n\); (2) if \(L\) has two fixed points then \(n\) is even and \(L\simeq\mathbf{2}^n\); (3) if \(L\) has a unique fixed point then \(L\) contains \(\mathbf{2}^n\overline\oplus\text\textbf{2}^n\) as a full subalgebra.