A coalgebraic take on regular and $\omega$-regular behaviours

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is monad. The first part of the paper devoted to presenting construction monad suitable modelling (in)finite behaviour. second focuses on concepts (coalgebraic) automaton its ($\omega$-) end Kleene-type theorems regular input. framework instantiated non-deterministic (B\"uchi) automata, tree automata probabilistic automata.