On the structure theory of partial automaton semigroups

D'Angeli, Daniele; Rodaro, Emanuele; Wächter, Jan Philipp

doi:10.1007/s00233-020-10114-5

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…We will show that the monogenic inverse subsemigroup S generated by f is free. The proof is inspired by [2,Example 23].…”

Section: Monogenic Free Inverse Semigroupsmentioning

confidence: 99%

“…Inverse semigroups of partial automorphisms of (infinite) regular rooted trees naturally generalize the important and intensively investigated notion of automorphism groups of regular rooted trees [4]. One can define them by using partial invertible automata over finite alphabets [14,12,2]. Alternatively, they can be regarded as inverse limits of inverse semigroups of partial automorphisms of finite regular rooted trees.…”

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confidence: 99%

“…Then we prove a criterion when the inverse semigroup generated by a one-to-one mapping is monogenic free inverse. In Section 3 we recall required definitions on automorphisms and partial automorphism of regular rooted trees; for details, the reader can refer to [4,12,2]. Then we show how a level transitive regular rooted tree automorphism satisfying certain conditions can be extended to a partial automorphism of another regular rooted tree so that the inverse semigroup generated by this extended partial automorphism is monogenic free inverse.…”

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confidence: 99%

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Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

Kochubinska,

Oliynyk

2024

Mat. Stud.

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For a one-to-one partial mapping on an infinite set, we present a criterion in terms of its cycle-chain decomposition that the inverse subsemigroup generated by this mapping is monogenic free inverse. We also give a sufficient condition for a regular rooted tree partial automorphism to extend to a partial automorphism of another regular rooted tree so that the inverse semigroup gene\-ra\-ted by this extended partial automorphism is monogenic free inverse. The extension procedure we develop is then applied to $n$-ary adding machines.

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“…We will show that the monogenic inverse subsemigroup S generated by f is free. The proof is inspired by [2,Example 23].…”

Section: Monogenic Free Inverse Semigroupsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

Kochubinska,

Oliynyk

2024

Mat. Stud.

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An Automaton Group with PSPACE-Complete Word Problem

Wächter

Weiß

2022

Theory Comput Syst

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We construct an automaton group with a -complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely -complete, compressed word problem and acts over a binary alphabet. Thus, it is optimal in terms of the alphabet size. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D’Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a -complete word problem and the second one is to utilize a construction used by Barrington to simulate Boolean circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

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Orbit expandability of automaton semigroups and groups

D'Angeli

Rodaro

Wächter

2020

Theoretical Computer Science

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We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which ω-words admit infinite orbits: for such a word, every prefix is expandable.In this paper, we show that, on input of a word u, an automaton T and a number k, it is decidable to check whether u is k-expandable with respect to the action of T . In fact, this can be done in exponential nondeterministic space. From this nondeterministic algorithm, we obtain a bound on the length of a potential orbit-increasing suffix x. Moreover, we investigate the situation if the automaton is invertible and generates a group. In this case, we give an algebraic characterization for the expandability of a word based on its shifted stabilizer. We also give a more efficient algorithm to decide expandability of a word in the case of automaton groups, which allows us to improve the upper bound on the maximal orbit-increasing suffix length. Then, we investigate the situation for reversible (and complete) automata and obtain that every word is expandable with respect to these automata. Finally, we give a lower bound example for the length of an orbit-increasing suffix.

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On the structure theory of partial automaton semigroups

Cited by 5 publications

References 16 publications

Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

An Automaton Group with PSPACE-Complete Word Problem

Orbit expandability of automaton semigroups and groups

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