# String rewriting systems and associated finiteness conditions

McGlashan, S. (2002) String rewriting systems and associated finiteness conditions. PhD thesis, University of Glasgow.

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Printed Thesis Information: https://eleanor.lib.gla.ac.uk/record=b2068595

## Abstract

We begin with an introduction which describes the thesis in detail, and then a
preliminary chapter in which we discuss rewriting systems, associated complexes and
finiteness conditions. In particular, we recall the graph of derivations r and the 2-
complex V associated to any rewriting system, and the related geometric finiteness
conditions F DT and F HT. In §1.4 we give basic definitions and results about finite
complete rewriting systems, that is, rewriting systems which rewrite any word in a
finite number of steps to its normal form, the unique irreducible word in its congruence
class.
The main work of the thesis begins in Chapter 2 with some discussion of rewriting
systems for groups which are confluent on the congruence class containing the empty
word. In §2.1 we characterize groups admitting finite A-complete rewriting systems
as those with a A-Dehn presentation, and in §2.2 we give some examples of finite
rewriting systems for groups which are A-complete but not complete.
For the remainder of the thesis, we study how the properties of finite complete
rewriting systems which are discussed in the first chapter are mirrored in higher
dimensions. In Chapter 3 we extend the 2-complex V to form a new 3-complex VP,
and in Chapter 4 we define new finiteness conditions F DT2 and F HT2 based on the
homotopy and homology of this complex. In §4.4 we show that if a monoid admits a
finite complete rewriting system, then it is of type F DT2 •
The final chapter contains a discussion of alternative ways to define such higher
dimensional finiteness conditions. This leads to the introduction, in §5.2, of a variant
of the Guba-Sapir homotopy reduction system which can be associated to any co~­
plete rewriting system. This is a rewriting system operating on paths in r, and is
complete in the sense that it rewrites paths in a finite number of steps to a unique
"normal form" .

Item Type: Thesis (PhD) Doctoral Q Science > QA Mathematics > QA75 Electronic computers. Computer science College of Science and Engineering > School of Computing Science Pride, Prof. Steve J. 2002 Miss Louise Annan glathesis:2002-6190 Copyright of this thesis is held by the author. 12 Mar 2015 12:40 12 Mar 2015 12:41 https://theses.gla.ac.uk/id/eprint/6190