scholarly journals Groupoids and the algebra of rewriting in group presentations

2020 ◽  
Author(s):  
N. D. Gilbert ◽  
E. A. McDougall
Keyword(s):  
Reduced Form ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Group Presentations ◽  
Rewrite Rules ◽  
Fundamental Groupoid

Abstract Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.

scholarly journals Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

2021 ◽  
Vol 178 (3) ◽  
pp. 173-185
Author(s):  
Arthur Adinayev ◽  
Itamar Stein
Keyword(s):  
Isomorphism Problem ◽  
Complete Characterization ◽  
Hasse Diagram ◽  
Subgraph Isomorphism ◽  
Rewriting Systems ◽  
Rewriting System ◽  
String Rewriting ◽  
Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

REWRITING SYSTEMS IN ALTERNATING KNOT GROUPS

2006 ◽  
Vol 16 (04) ◽  
pp. 749-769 ◽  
Author(s):  
FABIENNE CHOURAQUI
Keyword(s):  
Word Problem ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Regular Position

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the augmented Dehn presentation of the knot group of a tame, prime, alternating knot in regular position, with a common projection has a finite and complete rewriting system. This provides an algorithm for solving the word problem with this presentation and we find an algorithm for solving the word problem with the Dehn presentation also.

COMPLETE REWRITING SYSTEMS FOR CODIFIED SUBMONOIDS

2005 ◽  
Vol 15 (02) ◽  
pp. 207-216
Author(s):  
ANTÓNIO MALHEIRO
Keyword(s):  
The Other ◽  
Maximal Subgroups ◽  
Rewriting Systems ◽  
Group Of Units ◽  
Rewriting System ◽  
Other Hand

Given a complete rewriting system R on X and a subset X0 of X+ satisfying certain conditions, we present a complete rewriting system for the submonoid of M(X;R) generated by X0. The obtained result will be applied to the group of units of a monoid satisfying H1 = D1. On the other hand we prove that all maximal subgroups of a monoid defined by a special rewriting system are isomorphic.

scholarly journals Subgroups of finite index in groups with finite complete rewriting systems

2000 ◽  
Vol 43 (1) ◽  
pp. 177-183 ◽  
Author(s):  
S. J. Pride ◽  
Jing Wang
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Rewriting Systems ◽  
Rewriting System

AbstractWe show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

REWRITING SYSTEMS AND ORDERINGS ON ARTIN MONOIDS

2007 ◽  
Vol 17 (01) ◽  
pp. 61-75 ◽  
Author(s):  
PATRICK BAHLS ◽  
TYLER SMITH
Keyword(s):  
Rewriting Systems ◽  
Large Type ◽  
Rewriting System

In this paper we introduce a complete rewriting system on any large-type Artin monoid. The rewriting system stems from a well-ordering defined by Burckel on the related class of braid monoids. As a consequence of our rewriting system's existence we will recover the fact that certain large-type monoids are well-orderable, and we will discern finer detail regarding the structure of this associated ordering.

MONOIDS PRESENTED BY REWRITING SYSTEMS AND AUTOMATIC STRUCTURES FOR THEIR SUBMONOIDS

2009 ◽  
Vol 19 (06) ◽  
pp. 771-790 ◽  
Author(s):  
ALAN J. CAIN
Keyword(s):  
New Technique ◽  
Finitely Generated ◽  
Automatic Structures ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Open Questions ◽  
Automatic Structure ◽  
A New Technique

This paper studies rr-, ℓr-, rℓ-, and ℓℓ-automatic structures for finitely generated submonoids of monoids presented by confluent rewriting system that are either finite and special or regular and monadic. A new technique is developed that uses an automaton to "translate" between words in the original rewriting system and words over the generators for the submonoid. This is applied to show that the submonoid inherits any notion of automatism possessed by the original monoid. Generalizations of results of Otto and Ruškuc are thus obtained: every finitely generated submonoid of a monoid presented by a confluent finite special rewriting system admits an automatic structure that is simultaneously rr-, ℓr-, rℓ-, and ℓℓ-automatic; and every finitely generated submonoid of a monoid presented by a confluent regular monadic rewriting system admits an automatic structure that is simultaneously rr- and ℓℓ-automatic. These structures are shown to be effectively computable. An algorithm is given to decide whether the monoid presented by a confluent monadic finite rewriting system is ℓr- or rℓ-automatic. Finally, these results are applied to yield answers to some hitherto open questions and to recover and generalize established results.

CONTEXT-FREE GRAMMARS WITH LINKED NONTERMINALS

2007 ◽  
Vol 18 (06) ◽  
pp. 1271-1282 ◽  
Author(s):  
ANDREAS KLEIN ◽  
MARTIN KUTRIB
Keyword(s):  
Generative Capacity ◽  
Closure Properties ◽  
Rewriting Systems ◽  
Rewriting System ◽  
The Family ◽  
Infinite Degree ◽  
New Type ◽  
Equivalent Systems ◽  
Context Free ◽  
Context Free Grammars

We introduce a new type of finite copying parallel rewriting system, i. e., grammars with linked nonterminals, which extend the generative capacity of context-free grammars. They can be thought of as having sentential forms where some instances of a nonterminal may be linked. The context-free-like productions replace a nonterminal together with its connected instances. New links are only established between symbols of the derived subforms. A natural limitation is to bound the degree of synchronous rewriting. We present an infinite degree hierarchy of separated language families with the property that degree one characterizes the family of regular and degree two the family of context-free languages. Furthermore, the hierarchy is a refinement of the known hierarchy of finite copying rewriting systems. Several closure properties known from equivalent systems are summarized.

String rewriting and homology of monoids

1997 ◽  
Vol 7 (3) ◽  
pp. 207-240 ◽  
Author(s):  
DANIEL E. COHEN
Keyword(s):  
The Other ◽  
Rewriting Systems ◽  
Rewriting System ◽  
String Rewriting

Results of Anick (1986), Squier (1987), Kobayashi (1990), Brown (1992b), and others, show that a monoid with a finite convergent rewriting system satisfies a homological condition known as FP∞.In this paper we give a simplified version of Brown's proof, which is conceptual, in contrast with the other proofs, which are computational.We also collect together a large number of results and examples of monoids and groups that satisfy FP∞ and others that do not. These may provide techniques for showing that various monoids do not have finite convergent rewriting systems, as well as explicit examples with which methods can be tested.

INFINITE WORDS AND CONFLUENT REWRITING SYSTEMS: ENDOMORPHISM EXTENSIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 443-490 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
PEDRO V. SILVA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Spaces ◽  
Topological Structures ◽  
Infinite Words ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Gromov Hyperbolic ◽  
Characterization Theorems

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endomorphism extensions, continuous extensions) are discussed. Characterization theorems and positive decidability results are proved for most cases.

IMPROVED NET REWRITING SYSTEMS–EXTENDED PETRI NETS SUPPORTING DYNAMIC CHANGES

2008 ◽  
Vol 17 (06) ◽  
pp. 1027-1052 ◽  
Author(s):  
JUN LI ◽  
XIANZHONG DAI ◽  
ZHENGDA MENG ◽  
LIBO XU
Keyword(s):  
Petri Nets ◽  
The State ◽  
Automated Manufacturing ◽  
Dynamic Changes ◽  
Manufacturing Cell ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Practical System ◽  
Extended Petri Nets

The existing extensions of Petri nets supporting dynamic changes cannot ensure preservation of desired properties of the modeled systems. This paper tries to address this problem. By extending the structure of the net rewriting system (NRS) and modifying the specification of its rewriting rule, we present improved net rewriting system (INRS). INRS enjoys four important properties, i.e., locality, transitivity, reflexivity, and reversibility. A practical system subject to dynamic changes can be modeled by an INRS, and its desired properties can be preserved with the state evolvement of the INRS. The property-preserving principles of INRS and their strict proofs are provided as well. Finally, a case study is provided to demonstrate the modeling of an automated manufacturing cell with changeable structure using an INRS. The result shows that the desired properties of the cell are still preserved despite its model evolvement.

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