Trace monoids with idempotent generators and measure-only quantum automata

In this paper, we analyze a model of 1-way quantum automaton where only measurements are allowed (MON-1qfa). The automaton works on a compatibility alphabet (Sigma, E) of observables and its probabilistic behavior is a formal series on the free partially commutative monoid FI(Sigma, E) with idempotent generators. We prove some properties of this class of formal series and we apply the results to analyze the class LMO(Sigma, E) of languages recognized by MON-1qfa's with isolated cut point. In particular, we prove that LMO(Sigma, E) is a boolean algebra of recognizable languages with finite variation, and that LMO(Sigma, E) is properly contained in the recognizable languages, with the exception of the trivial case of complete commutativity.