Failure detector class k has been defined in [18] as an extension to failure detector , and an algorithm has been given in [16] to solve k-set agreement using k in asynchronous message-passing systems. In this paper, we extend these previous work in two directions. First, we define two new classes of failure detectors k and k, which are new ways of extending , and show that they are equivalent to k. Class k is more flexible than k in that it does not require the outputs to stabilize eventually, while class k does not refer to other processes in its outputs. Second, we present a new algorithm that solves k-set agreement using k when a majority of processes do not crash. The algorithm is a faithful extension of the Paxos algorithm [12], and thus it inherits the efficiency, flexibility, and robustness of the Paxos algorithm. In particular, it has better message complexity than the algorithm in [16]. Both the new failure detectors and the new algorithm enrich our understanding of the k-set agreement problem.