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In this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscripts should not exceed two double-spaced type-written pages.
Let F+(X) be the set of words of positive length over a finite set X. By an automaton mapping (over (X,Y)) we understand a mapping of F+(X) into a finite set Y where |Y|?1). The family of all mappings over (X,Y) may be considered as an infinite automaton U having 2? states. U has at most 2^{2?} subautomata and at most 2? countable subautomata. We show that these bounds are actually attained.