On recognizable and rational formal power series

We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semi-ring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly di erent) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochma nski's theorems.