Abstract. The paper carries out set-theoretic analysis of the structure of attributed transition systems without hidden transitions. Partial operations of composition of histories and traces are proposed. It is shown that they can be used to parallelize the design of coverings of sets of histories and traces. Equivalence relations on the set of states are extracted. In terms of systems with distinguished initial and final states, as well as systems with distinguished initial states and sets of final limit sets of states, classes of safe and correct systems are defined. The algebra of such systems is proposed.

Keywords: attributed transition systems without hidden transitions and their compositions, safeness and correctness, the structure of sets of states, histories and traces.