In this paper we study transmission of information over a multiple access channel (MAC) with noiseless feedback. We formulate this problem as a decentralized stochastic control problem, the three controllers being the decoder and the two encoders who, in the presence of limited information about each other, decide what to transmit at each time instance, in order to jointly achieve a common goal. Our contribution is two-fold. First, we identify structural properties of the optimal communication system that result in considerable simplification of the encoding/decoding process. The derived structural properties make it possible to consider transmission schemes that are akin to the posterior-matching scheme (PMS) for the point-to-point channel. Since the optimal communication system has this structure, we need only restrict attention to the study of those simplified systems, even when the optimal one is not known. Second, the aforementioned structural results allow us to view the original MAC system as an equivalent point-to-point communication system over a Markov channel with perfect state observation and delayed state feedback. Based on this equivalence, we derive a single-letter expression for the capacity of the original channel.