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Superpositions of periodic dot screens are largely used in electronic imaging in the field of color printing. In such superpositions the interaction between the superposed layers may cause new structures to appear which did not exist in any of the original layers: macrostructures (also known as moiré patterns) and microstructures (also known as rosettes). While macrostructures are not always generated in the superposition (cf. moiré-free superpositions), microstructures exist practically in any superposition, except for the most trivial cases. In fact, even the macrostructures, whenever they occur, consist of variations in the microstructure of the superposition. In the present paper we investigate the microstructures that appear in the superposition of periodic structures and their properties. We also find the conditions on the superposed layers under which the microstructure of the superposition varies-or remains invariant-when individual layers in the superposition are laterally shifted with respect to each other