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In this paper, controllability, observability, and minimality of 2-D separable denominator systems (SDS's) are studied based on the reduced-dimensional decomposition proposed by the authors in a previous paper. These notions of an SDS are completely related to those of its ID decomposition pair. On the basis of these relations, several new necessary and sufficient conditions are given to examine these notions of an SDS. These conditions are much simpler than any conventional conditions. Moreover, using these relations, we prove that the basic system theoretical problems of constructing controllable (or observable) or minimal state-space realizations for a given 2-D separable denominator transfer function matrix can be changed into corresponding 1-D problems. Therefore, any techniques developed in 1-D system theory can be used to solve these 2-D problems.