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This paper extends the previous work on common positive definite solutions(CPDSs) to planar algebraic Riccati inequalities(ARIs) to those with arbitrary dimensions. The topological structure of the set of all positive definite solutions of an ARI is investigated. This leads to a necessary and sufficient condition for the existence of CPDSs to a set of Riccati inequalities. It also reveals that the solution set of ARIs is a positive cube in Rn, which arouses a new method to search the CPDS. Some examples of three-dimensional ARIs are presented to show the effectiveness of the proposed methods. Unlike linear matrix inequality(LMI) method, the computing collapse will not occur with the increase of the number of Riccati inequalities due to the fact that our approach handles the ARIs one by one rather than simultaneously.