We consider an infinite horizon optimal control problem for a continuous-time Markov chain X in a finite set I with noise-free partial observation. The observation process is defined as Y-t = h(X-t), t >= 0, where h is a given map defined on I. The observation is noise-free in the sense that the only source of randomness is the process X itself. The aim is to minimize a discounted cost functional and study the associated value function V. After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function v associated with the latter control problem and the original value function V. Then, we present two different characterizations of v (and indirectly of V): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control.