  We study a model of one-way quantum automaton where only measurement
operations are allowed (MOn-1qfa). We give an algebraic characterization of
LMO, showing that the syntactic monoids of the languages in LMO are exactly the
literal pseudovariety of J-trivial literally idempotent monoids, where J is the
Green's relation determined by two-sided ideals. We also prove that LMO
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.
