Commutative unary algebras with modular and distributive topology lattices

摘要

In the present paper, we describe commutative unary algebras with finitely many operations whose topology lattices are modular, distributive, or Boolean, respectively. Moreover, the classes of all modular, distributive, or Boolean lattices that are isomorphic to a topology lattice of some commutative unary algebras with finitely many operations are characterized. In particular, it is proved that an arbitrary modular (distributive) lattice L is isomorphic to the topology lattice of some commutative unary algebra with finitely many operations if and only if L is isomorphic to the lattice of subgroups of some finite abelian (cyclic) group.