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In this paper, the concept of a recursible multidimensional system is introduced. These systems are a generalization of causal one-dimensional (1-D) systems, but are not tied to any particular ordering. With regard to implementation, they occupy the same position of importance that causal systems occupy in the 1-D case. A framework for the characterization and structural description of recursible systems is presented. Included in this description are a theorem characterizing recursible systems and recursible multidimensional difference equations, a generalized signal flow graph formalism, a new state-space model, and an extended state-space representation which allows algorithms not describable by a state-space model to be described by a state-space-like framework. Examples are given to illustrate each of these concepts.