In this paper the algebraic Riccati equation (ARE) of the discrete-time linear-quadratic (LQ) optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability. With respect to an appropriate basis, the discrete-time algebraic Riccati equation (DARE) decomposes into a Lyapunov equation and an irreducible Riccati equation, The focus is on the Riccati part which amounts to studying a DARE where all unimodular modes are controllable, A bijection between positive semidefinite solutions and certain well-defined sets of F-invariant subspaces is established which, together with its inverse, is order reversing. As an application, issues concerning positive definite or strong solutions are clarified, Analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace.