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In this paper, we study the multi-decision decentralized control framework, introduced in the work of Chakib and Khoumsi (2008), within the context of C&P, D&A and C&PVD&A architectures. The new framework is based on the fact that each supervisor issues a tuple of so-called micro-decisions, instead of a single decision. For each architecture, we define the notion of m-coobservability, which is useful to characterize the class of achievable languages. Stronger and decidable varieties of m-coobservability are defined in order to cope with the problem of decomposing infinite languages encountered in verifying the m-coobservability. The decidability of the new varieties is ensured in the case of regular languages.