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.
In his paper [
In his paper [
Let
Now, to a decomposition
In the present contribution, we adapt Brylawski’s results to the theory of (string) rewriting systems which we think is the natural framework to deal with theoretical notions of decomposition. Moreover we extend previous works by allowing noncommutative and even partially commutative, decompositions. Our main result, Theorem
We warm the reader that this work is not a contribution to the theory of string rewriting systems but should only be considered as a use of this theory to provide a unified treatment of several phenomena of decompositions that seem different in appearance but which are actually quite similar (see Section 4.3). It is not our goal to prove convergence or confluence or other properties of the reduction rules we consider, and sometimes these properties are even assumed to hold. Our few results about rewriting systems are quite easy to check (nevertheless, for the sake of completeness their proofs are given) and may even be considered as obvious for specialists of the field of string rewriting systems, but the goal of this paper is to provide some theoretical explanations of some phenomena that are encountered by nonspecialists.
The categorical notions used in this contribution, that are not defined here, come from [
In what follows
Each of the categories
There are also obvious forgetful functors from
The forgetful functor from
The left adjoint of the forgetful functor from groups to monoids may be described as the (unique) solution of the following universal problem. Let
Other universal problems, which will play an important role in what follows, are the free partially commutative structures. These structures have been introduced in [
Let
It can be shown that there exists a unique object
We may note that
We may clarify the relations between the free partially commutative structures. Using universal properties, it is not difficult to check that
The monoid
To prove this lemma it is sufficient to check that
There is also an important relation between
Let
The set-theoretical mapping
Actually a result from [
Let
According to [
More generally, let
Under the previous assumptions, there is an arrow
Let
Let
According to Lemma
Let For all
Let
In this short section, as in the following, we adopt several notations and definitions from [
Let
Now, let us assume that
In what follows are considered (particular) string rewriting systems on free partially commutative semigroups. As explained in the end of Section
Let
In this paper we only consider this kind of rewriting systems that may be considered as really restricted but we warm the reader that the alphabets we have in mind may have rich structures: see for instance Section 4.3 where
From now on in this current Section 4.1, and only for this subsection, we assume that
We study some algebraic consequences of convergence of this alphabetic rewriting system on irreducible elements in the form of some lemmas and corollaries. The main result (Proposition
Let
Let us assume that
The map
Let
The semigroups
As introduced in Section 3.1, let
The fact that the rewriting system is alphabetic (Definition
This characterization of
If
Let us assume that
Forthcoming Proposition
The following lemma reveals the structure of free partially commutative semigroup of
The semigroup
Let
Let
Informally speaking, according to Definition
Let us assume that
Let
Since we will deal with the empty word, one needs to recall the following. According to Lemma
Let
Let
We are now in position to establish the main result of this paper.
Let
Let
Now, let us assume that
The group
This section is devoted to the presentation of several examples of Tutte-Grothendieck groups and universal invariants corresponding to convergent alphabetic rewriting systems. These examples come from the theory of graphs (Tutte polynomial), from algebra (Weyl algebra, and Poincaré-Birkhoff-Witt theorem) and from combinatorics (prefabs).
In its famous paper [
For any set
Let
Let
Let
In [
Let
As examples of (unique and very unique factorization) prefabs, one can cite the following two from [
Let
In this appendix are briefly presented the main definitions and results of Brylawski’s theory that are extended and clarified in this contribution.
Let for every for every
A partial ordered set of this kind is called a
Let