Tree automata are widely used in various contexts. They are closed under boolean operations and their emptiness problem decidable in polynomial time. Dag automata are natural extensions of tree automata, operating on dags instead of on trees; they can also be used for solving problems. Our purpose in this paper is to show that algebraically they behave differently: the class of dag automata is not closed under complementation, dag automata are not determinizable, their membership problem is NP-complete, the universality problem is undecidable, and the emptiness problem is NP-complete even for deterministic labeled dag automata.