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We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra A with a commutative subspace lattice is strictly cyclic iff Lat(A)± contains a finite number of atoms and each nonzero element of Lat(^)x contains an atom. This leads to a characterization of the «-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero'sdoi:10.1090/s0002-9947-1991-0989575-0 fatcat:parpwsvvyvbchlui4di6ta745q