A possible approach to studying behavioural equivalencess in labelled transition systems is that of characterizing them in terms of homomorphic transformations. This characterization permits relying on algebraic techniques for proving systems proprties and reduces equivalence checking of two systems to studying the reltionships among the elements of their structures. Different algebriac characterizations of bisimulation-based equivalences in terms of particular transition systems homomorphisms have been proposed in the literature. Here we show, by an example, that trace-based equivalences are not locally characterizable and thus that the above result cannot be extended to these equivalences. However, imilar results can be obtained if we confine ourselves to restricted classes of transition systems. Here, the algebraic characterizations of three well known decorated-trace equivalences for tree-like structurs are presented.
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