Consensus has been regarded as the fundamental problem that must be solved to implement a fault-tolerant distributed system. However, only a weaker problem than traditional consensus need be solved. We generalize the consensus problem to include both traditional consensus and this weaker version. A straightforward generalization of the Paxos consensus algorithm implements general consensus. The generalizations of consensus and of the Paxos algorithm require a mathematical detour de force into a type of object called a command-structure set. - - - ENGINEER'S ABSTRACT The state-machine approach to implementing a fault-tolerant distributed system involves reaching agreement on the sequence of system commands by executing a sequence of separate instances of a consensus algorithm. It can be shown that any fault-tolerant asynchronous consensus algorithm requires at least two message delays between the issuing of a command and when it can be executed. But even in the absence of faults, no algorithm can guarantee this fast an execution if two different commands are issued concurrently. We generalize the state-machine approach to involve reaching agreement on a partially ordered set of commands. By generalizing the Paxos consensus algorithm, we can implement a system in which concurrently issued commands can always be executed in two message delays if they are non-interfering, so it does not matter in which order those commands are executed. For many systems, concurrent commands are rarely interfering, so the generalized Paxos algorithm can be quite efficient. And command-structure sets are very simple.