(Extended Version) Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed ($\mon$). We give an algebraic characterization of $\lmo(\Sigma)$, showing that the syntactic monoids of the languages in $\lmo(\Sigma)$ are exactly the $J$-trivial literally idempotent syntactic monoids, where $J$ is the Green's relation determined by two-sided ideals. We also prove that $\lmo(\Sigma)$ coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to $\lmo(\Sigma)$ and to discuss definability issues in terms of the existential first-order logic $\Sigma_1[<]$ and the linear temporal logic without the next operator LTLWN.