The main purpose of this paper is to advance a general theory of direct sum decompositions of quasi-ordered sets and to show that certain significant decompositions of semigroups, automata and graphs can be carried out as direct sum decompositions of certain quasi-ordered sets associated to them. In this way we generalize certain results given by M. Petrich in [16] and [17], S. Bogdanović and M. Ćirić in [3], and [4], and M. Ćirić and S. Bogdanović in [5] and [6].