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This note is concerned with the stability of discrete-time dynamical systems employing saturation arithmetic in the state-space. A matrix measure is introduced so that it can administer the proximity evaluation of a matrix to the set of diagonal matrices, and the measure is utilized for making an additional condition to the Lyapunov-Stein matrix inequality. The solvability of the modified matrix inequality ensures not only the stability but also the absence of overflow oscillation under the state saturation arithmetic, and this approach has the advantage of being free from auxiliary parameters. As an application, the obtained result is applied to the stability analysis of two-dimensional dynamics. Numerical examples are given to illustrate the results.