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On The Word Problem for Free Products of Semigroups and Monoids
... We refer the reader to Nyberg-Brodda [61] for a full (rather straightforward) proof. Thus we have a combinatorial characterisation of super-AFLs in terms of rewriting systems. ...
... The result is obvious using Lemma 1.3 and the alternating products introduced by the author [61,63]. We provide a direct proof instead. ...
... The class of monoids with word problem in C = CF is closed under taking free products [13,Theorem 6.2]. Indeed, the author has shown that the same is true for any super-AFL C [61]. On the other hand, it is an open problem whether the same is true for the class of monoids with word problem in DCF. ...
A monoid is called special if it admits a presentation in which all defining relations are of the form w = 1 . Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.
... The terminology monadic ancestor property was introduced by the author in [96], and also appears in [95,97], but was treated implicitly already in [22,76], see especially [76,Lemma 3.4]. The idea of defining classes of languages via ancestry in rewriting systems is not new, and can be traced back at least to e.g. ...
... Because of the importance and utility of iterated substitution, Greibach [63] (later expanded in [64]) defined super-AFLs as a full AFL closed under nested iterated substitution (by [97,Proposition 2.2], this is equivalent to the definition of super-AFL as defined in §1.2). Not long after, the notion of a hyper-AFL was introduced, being any full AFL closed under iterated substitution [105,11]. ...
... In this section, we will generalise (in a fairly uncomplicated manner) the bipartisan ancestors introduced in [97] to polypartisan ancestors, and prove that this construction preserves certain language-theoretic properties of the languages it is applied to. We will use this construction to obtain the multiplication table for a monoid free product from the table for a semigroup free product. ...
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 e.g. by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids, or groups with a -multiplication table, where is any reversal-closed super-, in the sense of Greibach. In particular, we deduce that the free product of two groups with resp. indexed multiplication tables again has an resp. indexed multiplication table.
... The terminology monadic ancestor property was introduced by the author in [91], and also appears in [89,90], but was treated implicitly already in [21,73] [17,18]. EXAMPLE 1.2. ...
... That is, in the terminology of Section 1.8, we have only used the property that CF is a reversal-closed super-AFL. (Similarly, general statements involving reversal-closed super-AFLs appear as the main results in previous work by the author [89][90][91].) Hence, the main results of this article (Theorems A, B, 4.5, and their corollaries) remain valid if CF is replaced in the definition of word-hyperbolicity by any other reversal-closed super-AFL, such as IND or ET0L. We have chosen not to state our theorems in this general form to maintain clarity; there does not, at present, seem to be a great deal of interest in the language-theoretic properties of multiplication tables outside the case of CF (that is, hyperbolicity). ...
This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136 (3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, 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, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a -multiplication table, where is any reversal-closed super- . In particular, we deduce that the free product of two groups with with respect to indexed multiplication tables again has an with respect to an indexed multiplication table.
Elder, Kambites, and Ostheimer showed that if a finitely generated group H has word problem accepted by a G-automaton for an abelian group G, then H has an abelian subgroup of finite index. Their proof is, however, non-constructive in the sense that it is by contradiction: they proved that H must have a finite index abelian subgroup without constructing any finite index abelian subgroup of H. In addition, a part of their proof is in terms of geometric group theory, which makes it hard to read without knowledge of the field.We give a new, elementary, and in some sense more constructive proof of the theorem, in which we construct, from the abelian G-automaton accepting the word problem of H, a group homomorphism from a subgroup of G onto a finite index subgroup of H. Our method is purely combinatorial and contains no geometric arguments.Keywordsword problemG-automatonabelian group
A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups. Finally, we ask whether the group of units of a context-free monoid is always coherent.
A monoid is called special if it admits a presentation in which all defining relations are of the form w = 1 . Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.
Suppose that G is a finitely generated group and {\operatorname{WP}(G)} is the formal language of words defining the identity in G . We prove that if G is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then {\operatorname{WP}(G)} is not a multiple context-free language.
We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups.
For a full semi-AFL K, B(K) is defined as the family of languages generated by all K-extended basic macro grammars, while H(K) ⊆ B(K) is the smallest full hyper-AFL containing K; a full basic-AFL is a full AFL K such that B(K) = K (hence every full basic-AFL is a full hyper-AFL). For any full semi-AFL K, K is a full basic-AFL if and only if B(K) is substitution closed if and only if H(K) is a full basic-AFL. If K is not a full basic-AFL, then the smallest full basic-AFL containing K is the union of an infinite hierarchy of full hyper-AFLs. If K is a full principal basic-AFL (such as INDEX, the family of indexed languages), then the largest full AFL properly contained in K is a full basic-AFL. There is a full basic-AFL lying properly in between the smallest full basic-AFL and the largest full basic-AFL in INDEX.
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.
This book is an indispensable source for anyone with an interest in semigroup theory or whose research overlaps with this increasingly important and active field of mathematics. It clearly emphasizes "pure" semigroup theory, in particular the various classes of regular semigroups. More than 150 exercises, accompanied by relevant references to the literature, give pointers to areas of the subject not explicitly covered in the text.
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).
This paper considers the word problem for free inverse monoids of finite rank from a language theory perspective. It is shown that no free inverse monoid has context-free word problem; that the word problem of the free inverse monoid of rank 1 is both 2-context-free (an intersection of two context-free languages) and ET0L; that the co-word problem of the free inverse monoid of rank 1 is context-free; and that the word problem of a free inverse monoid of rank greater than 1 is not poly-context-free.
We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following.
(a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets.
(b) If G is a countably infinite Abelian group with finitely many elements of order 2 and β G is the Stone–Čech compactification of G as a discrete semigroup, then for every idempotent p ∈ β G \﹛0﹜, the subset ﹛ p ,− p ﹜ ⊂ β G generates algebraically the free product of one-element semigroups ﹛ p ﹜ and ﹛− p ﹜.
Subsemigroups and ideals of free products of semigroups are studied with respect to the properties of being finitely generated or finitely presented. It is proved that the free product of any two semigroups, at least one of which is nontrivial, contains a two-sided ideal which is not finitely generated as a semigroup, and also contains a subsemigroup which is finitely generated but not finitely presented. By way of contrast, in the free product of two trivial semigroups, every subsemigroup is finitely generated and finitely presented. Further, it is proved that an ideal of a free product of finitely presented semigroups, which is finitely generated as a semigroup, is also finitely presented. It is not known whether one-sided ideals of free products have the same property, but it is shown that they do when the free factors are free commutative.
The word problem is of fundamental interest in group theory and has been widely studied. One important connection between group theory and theoretical computer science has been the consideration of the word problem as a formal language; a pivotal result here is the classification by Muller and Schupp of groups with a context-free word problem. Duncan and Gilman have proposed a natural extension of the notion of the word problem as a formal language from groups to semigroups and the question as to which semigroups have a context-free word problem then arises. Whilst the depth of the Muller-Schupp result and its reliance on the geometrical structure of Cayley graphs of groups suggests that a generalization to semigroups could be very hard to obtain we have been able to prove some results about this intriguing class of semigroups.
We provide, for N ≥ 1, a solution to the word problem for the semigroup S having the presentation { a, b | a = aba N b }. For N = 2, this is example 7 in Howie and Pride [1]. Since solving the word problem for { a, b | a = aba N b } is equivalent to solving the word problem for { a, b|a = ba N ba }, the result here should be regarded as a special case of a more general result of Oganesyan [2], who solves the word problem for { a, b|a = bA }, A arbitrary.
Let a semigroup A be given by generators a 1 , a 2 , … , a d and defining relations u 1 = v 1 , u 2 = v 2 , … , u e = v e between these generators, the u i , v i being words in the generators. We then have a presentation of A , and write
The same generators with the same relations can also be interpreted as the presentation of a group, for which we write
Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.(Received January 16 1985)(Revised May 10 1985)
In semigroup theory, the notion of semiband which generalises that of band, may play an interesting role both in pure algebraic theory and in computing and programming theory.
A relatively complete study of the structure of semiband of type two is given together with a complete listing of their isomorphism class.
We are much indebted to G.J. Lallement for suggesting improvements in our presentation and for communicating his computations concerning semibands of type two.
In this paper, we give a survey of various connections between the theories of group presentations and formal languages, concentrating on groups in which the word problem is a regular, one-counter or context-free language. We also prove some new results on one-counter groups.
A new type of grammar for generating formal languages, called an indexed grammar, is presented. An indexed grammar is an extension of a context-free grammar, and the class of languages generated by indexed grammars has closure properties and decidability results similar to those for context-free languages. The class of languages generated by indexed grammars properly includes all context-free languages and is a proper subset of the class of context-sensitive languages. Several subclasses of indexed grammars generate interesting classes of languages.
In this chapter we introduce the string-rewriting systems and study their basic properties. Such systems are the primary subject of this work. We provide formal definitions of string-rewriting systems and their induced reduction relations and Thue congruences. Some of the basic ideas that occur in the study of term-rewriting systems are considered. We rely on Section 1.4 for basic definitions and notation for strings, and we rely on Section 1.1 for basic definitions and results on notions such as reduction, confluence, the Church-Rosser property, and so forth.
Confluent and other types of finite Thue systems are studied. Sufficient conditions are developed for every congruence class and every finite union of congruence classes defined by such a system to be a deterministic context-free language. It is shown that the word problem for Church-Rosser systems is decidable in linear time.
Nous examinons les groupes finis, ou qui sont extensions finies du groupe Z. Nous donnons plusieurs caracterisations de cette classe, soit en termes de langages formels, soit en termes de la theorie des automates, et nous montrons que, du point de vue des langages formels, ces groupes forment la classe la plus importante entre les groupes context-free et les groupes finis
The concept of an automaton group generalizes easily to semigroups, and the systematic study of this area is beginning. This paper aims to contribute to that study. The basic theory of automaton semigroups is briefly reviewed. Various natural semigroups are shown to arise as automaton semigroups. The interaction of certain semigroup constructions with the class of automaton semigroups is studied. Semigroups arising from Cayley automata are investigated. Various open problems and areas for further research are suggested.
Certain infinite Thue systems over a finite alphabet are studied, in particular, systems S⊆∑∗×(∑∪{e}) such that for each aϵ∑∪{e}, the set {u| (u,a)ϵS} is a context-freelanguage. The syntactic structure of sets of ancestors and sets of descendants is considered, as well as that of unions of congruence classes, taken over (infinite) context-free languages or regular sets. The common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable (even for finite systems). The word problem for confluent systems of this type is shown to be tractable. The question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation.
We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.
A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then S∞(L, the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in S∞ (L). The index languages form a superAFL which is not the least superAFL containing the one way stack languages. If L has a decidable emptiness problem, so does S∞ (L). If Ds is an AFA, L=L (Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L (Dw) = S∞(L) if and only if Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested.
String-rewriting systems, Texts and Monographs in Computer Science
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On the word problem for compressible monoids
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Word Problem Languages for Free Inverse Monoids, Descriptional Complexity of Formal Systems
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