Groups and semigroups: connections and contrasts

Meakin, John

doi:10.1017/cbo9780511721205.002

Cited by 17 publications

(27 citation statements)

References 136 publications

(169 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The notion of unambiguous congruence defined below allows us to transform any graph into its greatest unambiguous image. In group theory, this generalizes the notion of Stallings foldings [29].…”

Section: Definition 4 (Induced Categories)mentioning

confidence: 79%

“…The construction described above is a generalization of what is known in algebra as Stallings folding [29]. Observe that with G = V, {E a } a∈V , the least unambiguous congruence G equals the least fixpoint of the mapping F :…”

Section: Lemma 7 (Maximal Unambiguous Image)mentioning

confidence: 99%

“…This yields to our main decidability result (Theorem 32). In some sense, our proposal amounts to combining concepts and results arising from the theory of inverse semigroups [26,29] with graph transformation approaches [27,13,6,9].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Inverse Monoids of Higher-Dimensional Strings

Janin

2015

Theoretical Aspects of Computing - ICTAC 2015

View full text Add to dashboard Cite

Abstract.Halfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph morphism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Finally, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory.

show abstract

Section: Definition 4 (Induced Categories)mentioning

confidence: 79%

Section: Lemma 7 (Maximal Unambiguous Image)mentioning

confidence: 99%

See 1 more Smart Citation

Inverse Monoids of Higher-Dimensional Strings

Janin

2015

Theoretical Aspects of Computing - ICTAC 2015

View full text Add to dashboard Cite

show abstract

“…This result motivated subsequent work investigating the question of whether all one-relator inverse monoids of the form Inv A | w = 1 have decidable word problem. This problem has now been shown to have a positive answer in several cases including when w is: an idempotent word [30], a sparse word [12], a one-relator surface group relation, a Baumslag-Solitar relation, or a relation of Adjan type; see [15,29] and [32,Section 7]. Here w ∈ (A ∪ A −1 ) * is called an idempotent word if it freely reduces to the identity element in the free group F G(A).…”

Section: Introductionmentioning

confidence: 99%

“…There are several places in the literature where it is mentioned that the problem of whether inverse monoids of the form Inv A|w = 1 have decidable word problem remains unsolved; see e.g. [28,Section 2.3], [12,32]. The first place this question appears in the literature is in the paper [31] of Margolis, Meakin and Stephen.…”

Section: Introductionmentioning

confidence: 99%

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

Gray

2019

Invent. math.

View full text Add to dashboard Cite

We prove the following results: (1) There is a one-relator inverse monoid Inv A | w = 1 with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg from 2008, we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid Inv A|w = 1 with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis, Meakin and Stephen in 1987.

show abstract

Walking Automata in Free Inverse Monoids

Janin

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Walking automata, be they running over words, trees or even graphs, possibly extended with pebbles that can be dropped and lifted on vertices, have long been defined and studied in Computer Science. However, questions concerning walking automata are surprisingly complex to solve. In this paper, we study a generic notion of walking automata over graphs whose semantics naturally lays within inverse semigroup theory. Then, from the simplest notion of walking automata on birooted trees, that is, elements of free inverse monoids, to the more general cases of walking automata on birooted finite subgraphs of Cayley's graphs of groups, that is, elements of free E-unitary inverse monoids, we provide a robust algebraic framework in which various classes of recognizable or regular languages of birooted graphs can uniformly be defined and related one with the other. 1 Introduction General context. Walking automata, be they running over words, trees or even graphs, possibly extended with pebbles, have long been defined and studied in Computer Science[7,8]. For instance, tree walking automata with pebbles have been an important subject of study the last decades since they are natural abstract models of machine for XML query languages such as XPATH, or XML transformation languages such as XSL [6]. Although based on well studied computation models: finite state machines or pushdown automata, questions about walking automata are often surprisingly complex to solve and, to a lesser extent, quite dependent on such or such details in automata's definition. For instance, in the case of tree languages, bounding the number of pebbles an automaton leads to defining classes of recognizable languages. Various logical characterizations of these classes have been obtained [8] and difficult separation results have also been proved [2,3,1]. However, for separation results, proof arguments apply to the case of pebbles that are marked and visible [2,3], leaving open the cases of unmarked and/or invisible pebbles. Even though walking automata are sequential machines much like string automata, the classical algebraic tools that have been developed to study word automata are not easily applicable to tree walking automata. Despite numerous results, little is known about the underlying mathematical framework, say in algebra, that walking automata may induce. Contribution of the paper. In this paper, we initiate the development of an algebraic framework, within inverse semigroup theory, for walking automata. We provide a generic notion of automata walking on edge-labeled graphs. They act as some kind of observers of their input graphs much in the same way observational semantics has been defined in concurrency theory by Hennessy and Milner [17]. Unlike most classical definitions, we do not require walking automata to start and end in the same vertex, neither do we require the complete traversal of input structures. Moreover, the capacity given to a walking automaton to check or not the absence of an (incoming or outgoing) edge labeled by a given lett...

show abstract

Groups and semigroups: connections and contrasts

Cited by 17 publications

References 136 publications

Inverse Monoids of Higher-Dimensional Strings

Inverse Monoids of Higher-Dimensional Strings

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

Walking Automata in Free Inverse Monoids

Contact Info

Product

Resources

About