The Tutte-Grothendieck group of a convergent alphabetic rewriting system

TL;DR

This paper extends Brylawski's theory by replacing order relations with string rewriting systems and partial commutations, clarifying the relationship between semigroups and the Tutte-Grothendieck group, including non-convergent systems.

Contribution

It generalizes Brylawski's framework to include string rewriting and partial commutations, establishing the Tutte-Grothendieck group as a Grothendieck group completion of the semigroup.

Findings

The Tutte-Grothendieck group is the Grothendieck group completion of the semigroup.

Normal forms serve as universal invariants.

Universal constructions are possible even for non-convergent rewriting systems.

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 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 non commutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the later is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was even not mention 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 non convergent rewriting system, outside the scope of Brylawski's work.