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Generalized mutual exclusion constraints (GMECs) are a typical class of specifications for the supervisory control of discrete event systems in a Petri net formalism. This paper classifies the given constraints into elementary and dependent ones according to the linear dependency of their characteristic transition vectors that indicate the token count change of the concerned places. The dependent constraints are further divided into strongly and weakly dependent ones. A constraint is usually enforced by explicitly adding a monitor to a plant model. This research develops the conditions under which a dependent constraint is enforced due to the enforcement of the elementary constraints. The results developed in this paper are applied to the existing manufacturing-oriented Petri net classes. An algorithm is also proposed to identify a set of elementary constraints. Examples are used to demonstrate the proposed methods. Some potential extensions and applications are also discussed. This research improves the existing methods in computational efficiency and structure simplification of the supervisor, given a set of GMECs.