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We develop a new algebraic framework to reason about languages ofMazurkiewicz traces. This framework supports true concurrency and provides anon-trivial generalization of the wreath product operation to the tracesetting. A novel local wreath product principle has been established. The newframework is crucially used to propose a decomposition result for recognizabletrace languages, which is an analogue of the Krohn-Rhodes theorem. We provethis decomposition result in the special case of acyclic architectures andapply it to extend Kamp's theorem to this setting. We also introduce andanalyze distributed automata-theoretic operations called local and globalcascade products. Finally, we show that aperiodic trace languages can becharacterized using global cascade products of localized and distributedtwo-state reset automata.

Wreath/cascade products and related decomposition results for the concurrent setting of Mazurkiewicz traces (extended version)