When are dual Cayley automaton semigroups finite?
Victor Maltcev
本文证明了对有限半群$S$，对偶Cayley自动机半群$\mathbf{C^{\ast}}(S)$是有限的当且仅当$S$是$\mathcal{H}$ -平凡的且没有非平凡的右零子半群.
In this note we prove that, for a finite semigroup $S$, the dual Cayley automaton semigroup $\mathbf{C^{\ast}}(S)$ is finite if and only if $S$ is $\mathcal{H}$-trivial and has no non-trivial right zero subsemigroups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Cayley automaton semigroups
本文刻画了Cayley自动机半群是群、平凡、有限、自由、左零半群或右零半群的条件。
After reviewing automaton semigroups, we introduce Cayley Automata and the corresponding Cayley Automaton semigroups. We investigate which semigroups are isomorphic to their Cayley Automaton semigroup and give some results for special classes of semigroups. We answer a question posed by Cain relating to the dual construction.
We introduce a new class of semigroups arising from a restricted class of asynchronous automata. We call these semigroups "expanding automaton semigroups." We show that the class of synchronous automaton semigroups is strictly contained in the class of expanding automaton semigroups, and that the class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups. We investigate the dynamics of expanding automaton semigroups acting on regular rooted trees, and show that undecidability arises in these actions. We show that this class is not closed und…
This paper shows how to construct explicitly an automaton that generates an arbitrary numerical semigroup.
On the Cayley semigroup of a finite aperiodic semigroup
设$S$是有限半群.在本文中，我们介绍的功能$ϕ_s:S^* \to S^*$，首先定义的罗兹，由$ϕ_s([a_1,a_2 ,...,a_n]) = [sa_1,sa_1a_2,..., sa_1a_2 ... a_n]$。证明了若$S$是有限非周期半群，则由函数$\{ϕ_s\}_{s \in S}$生成的半群是有限非周期的.
This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under certain kinds of Rees matrix constructions, strong semilattices, and small extensions. Finally, we prove that no subsemigroup of $(\mathbb{N}, +)$ arises as an automaton semigroup. (Previously, $(\mathb…
We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated…
The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of 'free product followed by adjoining an identity'. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an autom…
We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup,namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication.The framework embraces many of the well-known classes of (automatic) semigroups: finite monoids, free semigroups, free commutative monoids, trace or divisibility monoids, braid or Artin-Tits or Krammer or Garside monoids, Baumslag-Solitar semigroups, etc.Like plactic monoids or Chinese monoids, some n…
In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for …
The Freeness Problem for Automaton Semigroups
本文解决了Grigorchuk等人提出的开放问题，证明自动机半群和幺半群的自由性问题不可判定，通过将Post对应问题编码到其中实现，且输入字母表可限制为恒定大小，还得到了其他半群概念的不可判定性结果。
The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity
本文扩展了自动机半群和幺半群的活动概念，证明了有界扩展活动的自动机对应的无限轨道ω-语言是确定性Büchi语言，从而解决了有界活动的完全自动机半群和幺半群的有限性判定问题。
This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We …
We show that an automaton group or semigroup is infinite if and only if it admits an $ω$-word (i. e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an $ω$ -word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. F…
We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at $ω$-word (i.e. right infinite words) with a finite orbit. We show that every automaton yielding an $ω$-word with a finite orbit already yields an ultimately…
Preserving self-similarity in free products of semigroups
本文研究自相似半群与自动机半群在自由积下的封闭性，利用基础半群间的同态和可计算构造证明自由积保持相关性质，并给出添加自由生成元等推论及限制例子。
In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been stu…
In this paper, we give an introduction to basic concepts of automaton semigroups. While we must note that this paper does not contain new results, it is focused on extended introduction in the subject and detailed examples.
We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable when the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in t…
We prove that the semigroup generated by a reversible Mealy automaton contains a free subsemigroup of rank two if and only if it contains an element of infinite order.
We prove that the semigroup generated by a finite state Mealy automaton $\mathcal{A}=(Q,A,τ)$ is infinite if and only if there exists some right-infinite word in the alphabet $A$ with infinite orbit.
We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given $x,y$, whether an Engel identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) is satisfied. This algorithm succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, nam…
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. Fro…
We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton semigroups.
We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata and to the decidability of freeness for semigroups generated by two-state invertible-reversible Mealy automata.
This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.
We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.
Girth of the Cayley graph and Cayley hash functions
本文综述已有Cayley哈希函数，利用半群元素乘法实现可增量更新的哈希，并选出一种安全性至今未被攻破的简单方案。
We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among homogeneous sequences, we single out those enjoying what we call self-similarity. It turns out that self-similar $2$-automatic sequences (viewed up to a permutation of their alphabet) are in bijection with many interesting objects, for example dessins d'enfants (covers of the Riemann sphere with three points removed). For any $p$ we show that, in the case of an …
We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information related to the whole automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be describe…
Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $τ: A^G \to A^G$ (where $G$ is any group and $A$ is any set), the the…
We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area.
In this paper we study ends of finitely generated semigroups. The ends we are working with are the ends of the undirected graphs of Cayley graphs of finitely generated semigroups. We prove that the number of ends is preserved for subsemigroups of finite Rees index, and prove the same result for finite Green index subsemigroups of cancellative semigroups.
Cayley hashing with cookies
本文提出“带饼干的凯莱哈希”方法，利用F_p上2x2矩阵作为半群，在保留凯莱哈希增量更新优势的同时，消除其易受第二原像攻击的隐患。
Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-…
Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups and automata lead to simplifications of the graphs on which the corresponding finite semigroups act. We show in particular that every finite semigroup can be finitely expanded so that the expansion acts on a labeled directed graph which resembles the right Cayley graph of a free Burnside semigroup in many respects.
Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $τ\colon A^S \to A^S$ is surjective.
Motivated by issues arising in computer science, we investigate the loop-free paths from the identity transformation and corresponding straight words in the Cayley graph of a finite transformation semigroup with a fixed generator set. Of special interest are words that permute a given subset of the state set. Certain such words, called minimal permutators, are shown to comprise a code, and the straight ones comprise a finite code. Thus, words that permute a given subset are uniquely factorizable as products of the subset's minimal permutators, and these can be further reduced to straight minim…
Membership and Conjugacy in Inverse Semigroups
本文研究了有限逆半群的成员与共轭问题，通过簇参数化分析其复杂性，建立了部分双射和凯莱表模型下的二分定理，并应用于自动机等问题，得出PSPACE完全或NC/NP等复杂性结果。
In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the De Bruijn automata into Mealy aut…
Efficient Compression in Semigroups
本文研究了有限半群伪变种中允许高效压缩的特性，改进了直线程序的界限，并证明了所有可解群的成员问题属于FOLL。
We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$…
In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If $U$ is any semigroup, and $A$ is a subset of $U$, then we denote by $\langle A\rangle$ the least subsemigroup of $U$ containing $A$. If $B$ is any other subset of $U$, then, roughly speakin…
Closed Image Characterizations of Locally Finite Groups via Cellular Automata
证明群G是局部有限的当且仅当每个线性细胞自动机V^G→V^G在prodiscrete拓扑下具有闭图像，解决了Ceccherini-Silberstein和Coornaert的开放问题7。
For any group $G$ and any set $A$, a cellular automaton (CA) is a transformation of the configuration space $A^G$ defined via a finite memory set and a local function. Let $\text{CA}(G;A)$ be the monoid of all CA over $A^G$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element $τ\in \text{CA}(G;A)$ is von Neumann regular (or simply regular) if there exists $σ\in \text{CA}(G;A)$ such that $τ\circ σ\circ τ= τ$ and $σ\circ τ\circ σ= σ$, where $\circ$ is the composition of functions. Such an element $σ$ is called a generalised i…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。