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- W4309434561 abstract "The set of synchronizing words of a given $n$-state automaton forms a regular language recognizable by an automaton with $2^n - n$ states. The size of a recognizing automaton for the set of synchronizing words is linked to computational problems related to synchronization and to the length of synchronizing words. Hence, it is natural to investigate synchronizing automata extremal with this property, i.e., such that the minimal deterministic automaton for the set of synchronizing words has $2^n - n$ states. The sync-maximal permutation groups have been introduced in [{sc S. Hoffmann}, Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words, LATA 2021] by stipulating that an associated automaton to the group and a non-permutation has this extremal property. The definition is in analogy with the synchronizing groups and analog to a characterization of primitivity obtained in the mentioned work. The precise relation to other classes of groups was mentioned as an open problem. Here, we solve this open problem by showing that the sync-maximal groups are precisely the primitive groups. Our result gives a new characterization of the primitive groups. Lastly, we explore an alternative and stronger definition than sync-maximality." @default.
- W4309434561 created "2022-11-27" @default.
- W4309434561 creator A5015210496 @default.
- W4309434561 date "2021-11-26" @default.
- W4309434561 modified "2023-09-29" @default.
- W4309434561 title "Sync-Maximal Permutation Groups Equal Primitive Permutation Groups" @default.
- W4309434561 doi "https://doi.org/10.48550/arxiv.2111.13527" @default.
- W4309434561 hasPublicationYear "2021" @default.
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