Behavioural equivalences of labelled transition systems are characterized in terms of homomorphic transformations. This permits relying on algebraic techniques for proving systems properties and reduces equivalence checking of two systems to studying the relationships among the elements of their structures. Different algebraic characterizations of bisimulation-based equivalences in terms of particular transition system homomorphisms have been proposed in the literature. Here, it is shown that trace and decorated trace equivalences can neither be characterized in terms of transition system homomorphisms, nor be defined locally, i.e., only in terms of action sequences of bounded length and of root-preserving maps. However, results similar to those for bisimulation can be obtained for restricted classes of transition systems. For tree-like systems, we present the algebraic characterizations of trace equivalence and of three well-known decorated trace equivalences, namely ready, ready trace equivalence and failure.
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