In this paper, we explore some issues relevant to fault monitoring in discrete event systems modeled by partially observed LPNs with, possibly indistinguishable observable events and acyclic unobservable subnet. Firstly, we address the (offline) fault-prognosability analysis problem. Subsequently, we tackle the two online problems of K-step fault prognosis and K-step predictive diagnosis. We propose algebraic formulations and solutions to these problems. Namely, a necessary and sufficient condition for fault-prognosability is established based on solving an integer optimization problem. The proposed approach is applicable for bounded Petri nets. As for the K-step prognosis and K-step predictive diagnosis, algebraic approaches based on state estimation on a sliding horizon are elaborated to produce relevant verdicts. The established results for K-step prognosis and K-step predictive diagnosis are applicable for both bounded and unbounded Petri nets.