The Tutte-Grothendieck group of an alphabetic rewriting system

Laurent Poinsot (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

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.

Mots clés: Semi-Thue system; semigroup; free partially commutative structure; Grothendieck group completion; universal invariant

Domaines: Mathématique discrète [cs.DM]; Théorie des groupes [math.GR]

Laurent Poinsot. The Tutte-Grothendieck group of an alphabetic rewriting system. ISRN Combinatorics, 2013, 2013, pp.1-11. ⟨10.1155/2013/574578⟩. ⟨hal-00655414⟩