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17
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The tech ..."
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Cited by 15 (5 self)
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The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
Detecting quasiconvexity: algorithmic aspects, from: “Geometric and
 Computational Perspectives on Infinite Groups”, Gilbert Baumslag et al (editors), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25
, 1995
"... Abstract. The main result of this paper states that for any group G with an automatic structure L with unique representatives one can construct a uniform partial algorithm which detects Lrational subgroups and gives their preimages in L. This provides a practical, not just theoretical, procedure fo ..."
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Cited by 13 (10 self)
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Abstract. The main result of this paper states that for any group G with an automatic structure L with unique representatives one can construct a uniform partial algorithm which detects Lrational subgroups and gives their preimages in L. This provides a practical, not just theoretical, procedure for solving the occurrence problem for such subgroups. 1. Generalized word problem and rational structures on groups The goal of this paper is to highlight connections between the theory of automatic groups and the generalized word problem and to demonstrate certain additional advantages of the class of automatic groups over the class of combable groups. We assume that the reader is familiar with the theory of automatic groups, regular languages and combable groups. Although some of the important definitions will be given, the reader is referred to [ECHLPT] for further details. A good overview of the theory of automatic groups can also be found in [BGSS]. We take for granted some basic facts about word hyperbolic groups and their connections with the theory of automatic groups. Here our main references are [Gr], [ABCFLMSS], [ECHLPT], [BGSS] and [GS]. An important discussion about combable groups can also be found in [A], [AB] and [N]. The author is grateful to the referee for greatly simplifying the proof of Proposition 1 and to
Relating rewriting techniques on monoids and rings: Congruences on monoids and ideals in monoid rings
 Theoretical Computer Science
, 1998
"... A first explicit connection between finitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution to the word problem in commutative monoids by deciding the ideal membership problem. The aim of this paper is to show in a similar fashion how cong ..."
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Cited by 9 (2 self)
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A first explicit connection between finitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution to the word problem in commutative monoids by deciding the ideal membership problem. The aim of this paper is to show in a similar fashion how congruences on monoids and groups can be characterized by ideals in respective monoid and group rings. These characterizations enable to transfer well known results from the theory of string rewriting systems for presenting monoids and groups to the algebraic setting of subalgebras and ideals in monoid respectively group rings. Moreover, natural onesided congruences defined by subgroups of a group are connected to onesided ideals in the respective group ring and hence the subgroup problem and the ideal membership problem are directly related. For several classes of finitely presented groups we show explicitly how Gröbner basis methods are related to existing solutions of the subgroup problem by rewriting methods. For the case of general monoids and submonoids weaker results are presented. In fact it becomes clear that string rewriting methods for monoids and groups can be lifted in a natural fashion to define reduction relations in monoid and group rings.
Properties of Monoids That Are Presented By Finite Convergent StringRewriting Systems  a Survey
, 1997
"... In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the rela ..."
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Cited by 6 (5 self)
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In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the relationships between them. 1 Introduction Stringrewriting systems, also known as semiThue systems, have played a major role in the development of theoretical computer science. On the one hand, they give a calculus that is equivalent to that of the Turing machine (see, e.g., [Dav58]), and in this way they capture the notion of `effective computability' that is central to computer science. On the other hand, in the phrasestructure grammars introduced by N. Chomsky they are used as sets of productions, which form the essential part of these grammars [HoUl79]. In this way stringrewriting systems are at the very heart of formal language theory. Finally, they are also used in combinatorial semig...
4Engel groups are locally nilpotent
 Internat. J. Algebra and Comput
, 2005
"... Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations. ..."
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Cited by 4 (1 self)
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Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.
Some Challenging Group Presentations
 Journal of the Australian Mathematical Society (Series A
, 1999
"... We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published d ..."
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Cited by 4 (2 self)
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We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with soluble length seven. The tools we use are coset enumeration and KnuthBendix rewriting, which are wellestablished as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance. 1. Introduction The examples we consider are deficiency zero presentations; that is, presentations with the same number of generators as defining relations. Thus the presentations themselves are relatively concise, suggesting that they may be challenging for computational tools. The first example, in Section 3, is a presentation proposed by Malcolm Wicks, which turns out to be a diffi...
A transducer approach to Coxeter groups
 J. Symbolic Computation
, 1999
"... this paper.) In contrast, we would like to show here that once the exchange condition is known, all the computations in these groups can be explicitly handled at the combinatorial level, and point out in particular how parabolic decompositions appear naturally in these questions. This is particularl ..."
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Cited by 2 (1 self)
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this paper.) In contrast, we would like to show here that once the exchange condition is known, all the computations in these groups can be explicitly handled at the combinatorial level, and point out in particular how parabolic decompositions appear naturally in these questions. This is particularly efficient in the case of finite groups, where we obtain a cascade of very small transducers (cf. sect. 3) handling the main "word processing" problems that one would like to deal with. In addition, we show how parabolic decompositions lead to a very efficient determination of the Bruhat order  a further indication of their relevance in computational questions. On a theoretical level, the results that we use are due to Deodhar ([3], [4]); our contribution has been the realization of their practical value in terms of computer implementations. We also present a general algorithm for finding the normal form (cf. sect. 1) of an arbitrary word in a general Coxeter group. We did not attempt to analyze the complexity of this algorithm in general, but it certainly becomes very efficient when used for the construction of the transducer tables by induction on the length; once this is done, the general algorithm is not needed any more. Of course the transducer tables could also have been constructed using, for instance, a geometric realization of the group, but our algorithm has the advantages of simplicity, of handling the noncristallographic cases with equal ease, and may also yield some additional information on the structure of normal forms. It is natural in this setting to ask about the KnuthBendix relations for the presentation (cf. sect. 4). It turns out that for the finite Coxeter groups they can be readily read off from our transducer tables. We have indicated them in ord...
Automatic groups and string rewriting, H
 909, Lecture Notes in Comput. Sci
, 1995
"... In recent years developments in geometric topology have led to a remarkable interplay between group theory, geometry, and the theory of automata and formal languages. One of the best known result of this interplay has been the ..."
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Cited by 1 (0 self)
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In recent years developments in geometric topology have led to a remarkable interplay between group theory, geometry, and the theory of automata and formal languages. One of the best known result of this interplay has been the