The study of the trace monoid was initiated by A. Mazurkiewicz in 1977, and it was later developed, motivated by its interpretation as a model to describe the behaviour of concurrent systems. When a partial commutation (often called independence relation) is fixed over the letters of an alphabet, stating what letters can commute, a trace can be identified with an equivalence class of words that can be obtained from one another by switching successively some consecutive independent letters.

In the first part of the talk we propose an extension of the notion of local testability from words to traces. A locally testable trace language is defined as a union of classes of a congruence of a finite index, which preserves the prefixes, the suffixes and the factors. We give a set-theoretic characterization of this family of trace languages, and we analyze what it become in the extreme cases of the free monoid and of the free commutative monoid.
In the second part we present a conjecture linking partial commutations and group languages, and we recall the results already known about it.