Table of Contents
ISRN Combinatorics
Volume 2013 (2013), Article ID 574578, 11 pages
http://dx.doi.org/10.1155/2013/574578
Research Article

The Tutte-Grothendieck Group of an Alphabetic Rewriting System

LIPN-UMR CNRS 7030, Université Paris Nord XIII, 99 avenue J.-B. Clément, 93430 Villetaneuse, France

Received 2 May 2012; Accepted 13 June 2012

Academic Editors: A. P. Godbole, M.-J. Jou, and B. Taeri

Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski (1972) in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski’s theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between noncommutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the latter is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was not even mentioned by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a nonconvergent rewriting system, outside the scope of Brylawski’s work.