Pomset languages recognized by higher-dimensional automata are regular iff inverse images of functors to finite categories and FO-definable iff recognized by aperiodic categories, generalizing McNaughton-Papert.
10 Uli Fahrenberg and Krzysztof Ziemianski
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Recognizable languages in free many-sorted algebras are regular under finitary assumptions.
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Algebraic Characterization of FO-definable Languages of Higher-Dimensional Automata
Pomset languages recognized by higher-dimensional automata are regular iff inverse images of functors to finite categories and FO-definable iff recognized by aperiodic categories, generalizing McNaughton-Papert.