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(eng): This paper introduces two types of preorders on the system of all non-empty sets of collections (i.e., the set of all decomposition systems) based on a fixed monotone measure mu. Each of them refines the previous two kinds of preorders of decomposition systems. By means of these two new preorders of decomposition systems we investigate the coincidences of decomposition integrals and that of superdecomposition integrals, respectively. The generalized integral equivalence theorem is shown in the general framework involving an ordered pair of decomposition systems. This generalized theorem includes as special cases all the previous results related to the coincidences among the Choquet integral, the concave (or convex) integral and the pan-integrals. Thus, a unified approach to the coincidences of several well-known decomposition and superdecomposition integrals is presented.