We consider a certain class of infinite monounary algebras that do not contain cyclic and meta-monogenic subalgebras. We show that a monounary algebra of this class is determined up to isomorphism by its semigroup of endomorphisms. It is also proved that, in the semigroup of endomorphisms of every monounary algebra of this class, there exists a densely imbedded ideal, whose cardinal number is equal to the order of this monounary algebra, and which also determines it.
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