Muhammad Inam
We show that all of the Schützenberger complexes of an Adian inverse semigroup are finite if the Schützenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).
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We prove that an inverse semigroup over an Adian presentation is E-unitary.
The word problem for a family of one relation Adian inverse semigroups
研究由表示$Inv\langle a,b|a=ba^nb\rangle$（$n\geq 1$）给出的Adian逆半群的字问题，并证明其可判定性。
The word problem for some classes of Adian inverse semigroups -- II
本文研究了Adian逆半群的词问题决定性，通过分析Schützenberger图的子图结构，证明了有限生成Adian逆半群的词问题可解性。
We show that if the Schützenberger graph of every positive word, that contains an $R$-word only once as it's subword, is finite over an Adain presentation $\langle X|u=v\rangle$, then the Schützenberger graph of every positive word is finite over the presentation $\langle X|u=v\rangle$. This enable us to solve the word problem ffor some classes of one relation Adian inverse semigroups.
This is an English translation of the thesis written by G. S. Makanin for the degree of Candidate of Physical and Mathematical Sciences (equivalent to a Ph.D.), originally submitted to the Steklov Mathematical Institute in 1966. The original language is Russian. The named supervisors are A. A. Markov and S. I. Adian.
We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if $\mathcal{C}$ is a reversal-closed super-$\operatorname{AFL}$, as defined by Greibach, then $M$ has word problem in $\mathcal{C}$ if and only if its compressed left monoid $L(M)$ has word problem in $\mathcal{C}$. As a special case, we may take $\mathcal{C}$ to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finall…
We prove that in an arbitrary semigroup without cycles, the problem of divisibility and, therefore, the word problem is solvable.
The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups.
Every semigroup which is a finite disjoint union of copies of the free mono- genic semigroup (natural numbers under addition) has soluble word prob- lem and soluble membership problem. Efficient algorithms are given for both problems.
We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigr…
The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of ma…
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…
The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by $\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into investigating the structure of semigroups of the form $\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for $\mathsf{IG…
Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered sets. We prove that such word problems are in fact equivalent to the problem of computing intersections of cosets of certain subgroups of direct products of maximal subgroups of the free idempotent generated semigroup in question, thus providing decidability of those word problems under group-theoretical assumptions related to the Howson property and the coset…
A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.
Let H be a pseudovariety of groups in which the kappa-word problem is decidable. Here, kappa denotes the canonical implicit signature, which consists of the multiplication and the (omega-1)-power. We prove that the kappa-word problem is also decidable over DRH, the pseudovariety of all finite semigroups whose regular R-classes lie in H. Further, we present a canonical form for elements in the free kappa-semigroup over DRH, based on the knowledge of a canonical form for elements in the free kappa-semigroup over H. This extends work of Almeida and Zeitoun on the pseudovariety of all finite R-tri…
With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, et…
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These…
We formulate and explain the extended Burrows-Wheeler transform of Mantaci et al from the viewpoint of permutations on a chain taken as a union of partial order-preserving mappings. In so doing we establish a link with syntactic semigroups of languages that are themselves cyclic semigroups. We apply the extended transform with a view to generating de Bruijn words through inverting the transform.
Motivated by the question of which completely regular semigroups have context-free word problem, we show that for certain classes of languages $\mathfrak{C}$(including context-free), every completely regular semigroup that is a union of finitely many finitely generated groups with word problem in $\mathfrak{C}$ also has word problem in $\mathfrak{C}$. We give an example to show that not all completely regular semigroups with context-free word problem can be so constructed.
This note proves a generalisation to inverse semigroups of Anisimov's theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- the set of all pairs of words over a generating set for the semigroup which both represent the same element.
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.
This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, includ…
In this paper we investigate the word problem of the free Burnside semigroup satisfying x^2=x^3 and having two generators. Elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique "canonical" representative for each equivalence class. We prove that overlap-free words and so-called almost overlap-free words (this notion is some generalization of the notion of overlap-free words) can serve as canonical representatives for corresponding equivalence classes. We show that such a word in a given class, if any, can be efficiently found.…
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…
We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When $\mathcal{C}$ is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in $\mathcal{C}$ also has word problem in $\mathcal{C}$. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of c…
This paper studies the classes of semigoups and monoids with context-free and deterministic context-free word problem. First, some examples are exhibited to clarify the relationship between these classes and their connection with the notions of word-hyperbolicity and automaticity. Second, a study is made of whether these classes are closed under applying certain semigroup constructions, including direct products and free products, or under regressing from the results of such constructions to the original semigroup(s) or monoid(s).
We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and algebraic properties of these semigroups.
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 …
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence also to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.
G. S. Tseytin's seven-relation semigroup with undecidable word problem
本文介绍 Tseytin 1958 年构造的具有不可判定字问题的七关系半群，梳理其思想历史、证明直观，并概述后续发展。
We show that the word problem for an amalgam $[S_1,S_2;U,ω_1,ω_2]$ of inverse semigroups may be undecidable even if we assume $S_1$ and $S_2$ (and therefore $U$) to have finite $\mathcal{R}$-classes and $ω_1,ω_2$ to be computable functions, interrupting a series of positive decidability results on the subject. This is achieved by encoding into an appropriate amalgam of inverse semigroups 2-counter machines with sufficient universality, and relating the nature of certain \sch graphs to sequences of computations in the machine.
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…
In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-pres…
This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the role that Cayley graphs have for group presentations or that Schützenberger graphs have for inverse monoid presentations. However, our graphs have considerable differences with the latter two, even though locally inverse semigroups generalize both groups and inverse semigroups. For example, the graphs introduced here are not `inverse word graphs'. Instead, th…
Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left- and by right multiplication. This gives rise to a partition of the complement of T in S, and to each equivalence class of this partition we naturally associate a relative Schutzenberger group. We show how generating sets for S may be used to obtain generating sets for T and the Schutzenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schutzenberger groups. These results are then used to show that several important properties are preser…
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…
Streaming algorithms for groups and semigroups
研究群与半群中字问题的确定性和随机性流算法，提出区分器概念，构造低空间复杂度的区分器，适用于线性半群和某些构造得到的半群，并给出下界证明部分半群不支持亚线性空间区分器。
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…
On Word Representations and Embeddings in Complex Matrices
研究词结构嵌入到低维复数矩阵半群中的方法，开发新的技术构建欧几里得比昂奇群的词表示，为分析2x2矩阵半群的基本决策问题提供框架。
The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which a…
The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalisation to inverse semigroups of Anisimov's Theorem for groups.
Finitely and non-finitely related words
本文利用链字、冠字词和漩涡字构造非有限相关幂零幺半群，揭示有限相关与非有限相关半群及簇之间的多种反例现象。
Small palindromic lengths in free groups and word equations with antimorphisms
本文研究了自由群中回文长度为2和3的词特性，并探讨了自由群中回文长度计算的复杂性。
Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $α\in \langle D \rangle$, let $\ell(D,α)$ be the minimal length of a word in $E(D)$ expressing $α$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,α)$,…
Word Equations: Pairs and their Makanin-Razborov Diagrams
本文提出用于编码自由半群方程组解集的 Makanin-Razborov 图，并展示其可用于分析自由半群的一阶理论片段。
The structure and density of $k$-product-free sets in the free semigroup
本文研究自由半群中k-乘积自由集的密度上界与极值结构，证明其上Banach密度至多1/ρ(k)，并确定多数k值下的极值结构及强k-乘积自由集的最大密度结构，同时将结果推广到自由群并证实相关猜想。
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。
An m-endomorphism of a free semigroup is an endomorphism that sends every generator to a word of length at most m. Two m-endomorphisms are combinatorially equivalent if they are conjugate under an automorphism of the semigroup. In this paper, we specialize an argument of N. G. de Bruijn to produce a formula for the number of combinatorial equivalence classes of m-endomorphisms on a rank-n semigroup. From this formula, we derive several little-known integer sequences.
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…
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…
This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure. Let $X$ be a finite set. We say that a primitive group $G$ on $X$ is {\em synchronizing} if $G$ together with any non-inve…
The finite basis problem for flat semirings $S_c(W)$, $M_c(W)$ and $M(W)$
本文研究平坦半环的有限基问题，完全分类了 $S_c(W)$、$M_c(W)$ 和 $M(W)$ 的有限基性质，并作为应用恢复了 $\flat(\mathbb{Z})$ 和 max-plus 半环的非有限基性。
We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Peter Mayr in 2013.
Stalactic, taiga, sylvester and Baxter monoids arise from the combinatorics of tableaux by identifying words over a fixed ordered alphabet whenever they produce the same tableau via some insertion algorithm. In this paper, three sufficient conditions under which semigroups are finitely based are given. By applying these sufficient conditions, it is shown that all stalactic and taiga monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities an…
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-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 PSP…
This paper makes a comprehensive survey of results relating to finite Rees index for semigroups. In particular, we survey of the state of knowledge on whether various finiteness properties (such as finite generation, finite presentability, automaticity, and hopficity) are inherited by finite Rees index subsemigroups and extensions. We survey other properties that are invariant under passing to finite Rees index subsemigroups and extensions, such as the cofinality and number of ends. We prove some new results: inheritance of word-hyperbolicity by finite Rees index subsemigroups, and inheritance…
In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether…
To any semigroup presentation $\mathcal{P}= \langle Σ\mid \mathcal{R} \rangle$ and base word $w \in Σ^+$ may be associated a nonpositively curved cube complex $S(\mathcal{P},w)$, called a Squier complex, whose underlying graph consists of the words of $Σ^+$ equal to $w$ modulo $\mathcal{P}$ where two such words are linked by an edge when one can be transformed into the other by applying a relation of $\mathcal{R}$. A group is a diagram group if it is the fundamental group of a Squier complex. In this paper, we describe hyperplanes in these cube complexes. As a first application, we determine e…
We present efficient computational solutions to the problems of checking equality, performing multiplication, and computing minimal representatives of elements of free bands. A band is any semigroup satisfying the identity $x ^ 2 \approx x$ and the free band $\operatorname{FB}(k)$ is the free object in the variety of $k$-generated bands. Radoszewski and Rytter developed a linear time algorithm for checking whether two words represent the same element of a free band. In this paper we describe an alternate linear time algorithm for checking the same problem. The algorithm we present utilises a r…
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