This paper studies left invertibility of discrete-time linear outputquantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a nite alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility (so also left invertibility for a full measure matrix set) for all linear systems with no eigenvalue of modulus one. Therefore we are able to check left invertibility of output-quantized linear systems for a full measure matrices set. Some examples are presented to show the application of the proposed method.