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In this paper, we clarify the notion of architecture in decentralized control, in order to investigate the realizability problem: given a discrete-event system, a desired behavior and an architecture for a decentralized control, can the desired behavior be achieved by decentralized controllers in accordance with the given architecture? We consider the problem for any mu-calculus definable behavior and for classic architectures from the literature. The method consists in compiling in a single formula both the desired behavior and the architecture. Applications of this approach are a single synthesis algorithm of decentralized controllers (with full observation) for the whole considered family of architectures, and the development of a convenient mathematical framework for a theory of decentralized control architectures.