Abstract
We search for the largest syntactic semigroup of a star-free language having n left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with n states.
We first introduce unitary semigroups generated by transformations that change only one state. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For \(n \geqslant 4\) there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups.
This work was supported by the Natural Sciences and Engineering Research Council of Canada grant No. OGP000087 and by Polish NCN grant DEC-2013/09/N/ST6/01194.
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Brzozowski, J., Szykuła, M. (2014). Large Aperiodic Semigroups. In: Holzer, M., Kutrib, M. (eds) Implementation and Application of Automata. CIAA 2014. Lecture Notes in Computer Science, vol 8587. Springer, Cham. https://doi.org/10.1007/978-3-319-08846-4_9
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DOI: https://doi.org/10.1007/978-3-319-08846-4_9
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