  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.
