Some properties of retract lattices of monounary algebras

Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ^∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ^∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ^∅(A, f) is complemented. In this case R ^∅(A, f) forms a boolean lattice.