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.