We investigate a new decentralized control architecture, called intersection-based architecture. We propose a decentralized control protocol under this architecture, called state-estimator-intersection-based protocol (SEI-protocol), where each local supervisor sends its state estimate to the fusion sites and these sites take the intersection of these estimates in order to make a control decision. The necessary and sufficient conditions for the achievability of the specification under this protocol is provided. This condition is termed as state-estimator-intersection-based coobservability (SEI-coobservability). A polynomial-time algorithm for the verification of SEI-coobservability is provided. We show that the languages that can be achieved under the SEI-protocol are incomparable with languages that can be achieved under existing architectures.