On finite lattices of topologies of commutative unary algebras

In this paper, we show that the lattice of topologies of a finite one-generated commutative unary algebra is isomorphic to the lattice of topologies of the characteristic semigroup of this algebra. With the use of this assertion, we give a characterisation of the class of all commutative unary algebras with linearly ordered lattices of topologies. It is proved that if the lattice of congruences or the lattice of topologies of a commutative unary algebra is finite, then the algebra itself is finite. Examples of infinite noncommutative unary algebras with finite lattices of topologies are given. It is proved that for an arbitrary functional signature containing at least one symbol with arity greater than 1 and for any integer $n\ge2$ there exists an infinite algebra of such signature whose lattice of topologies is linearly ordered and consists of $n$ elements.