The Tutte-Grothendieck Group of an Alphabetic Rewriting System

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.

1. Introduction

In his paper [1], Tutte took advantage of two natural operations on (finite multi) graphs (actually on isomorphism classes of multigraphs), deletion and contraction of an edge, in order to introduce the ring ℤ[𝑥,𝑦] and a polynomial in two commuting variables 𝑥,𝑦, also known by Whitney [2], unique up to isomorphism since solutions of a universal problem. This polynomial, since called the Tutte polynomial, is a graph invariant in at least two different meanings: first of all, it is defined on isomorphism classes, rather than on actual graphs, in such a way that two graphs with distinct Tutte polynomials are not isomorphic (a well-known functorial point of view), and, secondly, it is invariant with respect to a graph decomposition. Indeed, let 𝐺 be a graph, and let 𝑒 be an edge of 𝐺, which is not a loop (an edge with the same vertex as source and target) nor a bridge (an edge that connects two connected components of a graph). The edge contraction 𝐺/𝑒 of 𝐺 is the graph obtained by identifying the vertices source and target of 𝑒, and removing the edge 𝑒. We write 𝐺−𝑒 for the graph where the edge 𝑒 is merely removed; this operation is the edge deletion. Let us consider the graph 𝐺/𝑒+(𝐺−𝑒) (well-defined as isomorphic classes) which can be interpreted as a decomposition of 𝐺. Then, the Tutte polynomial 𝑡 is invariant with respect to this decomposition in the sense that 𝑡(𝐺)=𝑡(𝐺/𝑒)+𝑡(𝐺−𝑒). Moreover this decomposition eventually terminates with graphs with bridges and loops only as edges, and the choice of edges to decompose is irrelevant.

In his paper [3], Brylawski observed that the previous construction (and many others, for instance the Tutte polynomial for matroids) may be explained in terms of an elegant and unified categorical framework (namely, a universal problem of invariants). In brief, Brylawski considered an abstract notion of decomposition. Let 𝑋 be a set, and let ≤ be an order relation on (a part of) the free commutative semigroup 𝑋^⊕ (actually Brylawski considered multisets, nevertheless the choice is here made to deal with semigroups since they play a central role in this contribution), which satisfies a certain number of axioms that are quickly reviewed in informal terms below for the sake of completeness (the appendix contains a short review of Brylawski’s theory in mathematical terms but it may be skipped) and to show how natural is their translations in terms of rewriting systems.

Let 𝐷(𝑋) be a set of formal (finite) sums ∑_{1≤𝑖≤𝑘}𝑛_𝑖𝑥_𝑖, where 𝑥_𝑖∈𝑋, 𝑛_𝑖∈ℕ not all of them being zero (an element of the free commutative semigroup 𝑋^⊕ on 𝑋) partially ordered by ≤. If 𝑓,𝑔∈𝐷(𝑋) such that 𝑓≤𝑔, then we say that 𝑓  decomposes into 𝑔 or that 𝑔 is a decomposition of 𝑓. Therefore 𝐷(𝑋) is seen as a set of commutative decompositions. Elements of 𝑋 that belong to 𝐷(𝑋) are assumed to be minimal with respect to ≤. Elements of 𝑋∩𝐷(𝑋) that are maximal (and therefore incomparable since they are also minimal) are said to be irreducible. According to a second axiom satisfied by the order relation ≤, an element 𝑓∈𝐷(𝑋) cannot be decomposed further into any other element of 𝐷(𝑋) if and only if, 𝑓 is a finite linear combination, with nonnegative integers as coefficients, of incomparable elements, that is, if Irr(𝑋) is the set of all irreducibles, then 𝑓 is not decomposed into another element if and only if, 𝑓 is a formal (finite) sum of elements of Irr(𝑋) with nonnegative integers as coefficients. This property is similar to the notion of termination in rewriting systems. Two other properties (refinability and finiteness) on 𝐷(𝑋) ensure that every element of 𝑋 has one, and only one, “terminal” decomposition into irreducible elements. They are equivalent to convergence of a rewriting system. For instance, the order 𝐺<(𝐺/𝑒)+(𝐺−𝑒) on the free commutative semigroup generated by all (isomorphism classes of) finite graphs satisfies these axioms and properties.

Now, to a decomposition (𝐷(𝑋),≤) with the above properties may be attached a group in a universal way. A function 𝑓 from 𝑋 to an Abelian group 𝐺 is said to be invariant if for every 𝑥∈𝑋 such that 𝑥≤∑_{1≤𝑖≤𝑘}𝑛_𝑖𝑥_𝑖 (𝑥_𝑖∈𝑋, and 𝑛_𝑖∈ℕ), then 𝑓(𝑥)=𝑛_𝑖𝑓(𝑥_𝑖). Recall here that Tutte polynomial 𝑡 is invariant because 𝑡(𝐺)=𝑡(𝐺/𝑒)+𝑡(𝐺−𝑒). Brylawski proved the following theorem, which was his main result. There exist an Abelian group, called Tutte-Grothendieck group, and an invariant mapping 𝑡∶𝑋→𝐴, called universal Tutte-Grothendieck invariant, such that for every Abelian group 𝐺 and every invariant mapping 𝑓∶𝑋→𝐺, there exists a unique group homomorphism ℎ∶𝐴→𝐺 with ℎ∘𝑡=𝑓. In addition, 𝐴 is isomorphic to the free Abelian group with the irreducible elements as generators. In the classical context of graph theory, as expected, 𝐴 is the additive structure of ℤ[𝑥,𝑦] and 𝑡 is the Tutte polynomial. Many other decompositions enter in the scope of Brylawski’s theory (see his paper [3], examples and references therein).

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 16, similar to Brylawski’s main theorem, states the existence and uniqueness of a universal group and a universal invariant associated to some kind of string rewriting systems, even if they are not convergent (which is beyond the scope of Brylawski’s work). In case of convergence, we prove that the universal group under consideration is the free partially commutative group generated by irreducible letters, which is a generalization of the original result, and that the universal invariant is nothing else than the normal form function that maps an element to its normal form. We mention the fact that in this case, the universal group is proved to be the Grothendieck completion of a monoid (obtained from the semigroup subject to rewriting by free adjunction of an identity), which was not seen by Brylawski even if he called Tutte-Grothendieck his universal construction.

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.

2. Some Universal Constructions

The categorical notions used in this contribution, that are not defined here, come from [4]. This section is devoted to the presentation of Grothendieck group completion and free partially commutative structures which are used here after.

2.1. Basic Notions and Some Notations

In what follows 𝒮, ℳ, and 𝒢 denote the well-known categories of (small (“small” refers to some given fixed universe, see [4])) semigroups, monoids, and groups respectively, with their usual arrows (the so-called homomorphisms of semigroups, monoids, or groups).

Each of the categories 𝒮, ℳ, and 𝒢 has a free object freely generated by a given (small) set. In other terms their forgetful functors to the category of sets have a left adjoint. In what follows we denote by 𝑋⁺, 𝑋^∗, and 𝐹(𝑋), respectively, the free semigroup, monoid, group generated by 𝑋 (see [5]), and we identify 𝑋 as a subset of each of these algebraic structures. Note also that we denote by 𝑋^⊕ the free commutative semigroup on 𝑋.

There are also obvious forgetful functors from 𝒢 to ℳ, and from ℳ to 𝒮 (therefore also from 𝒢 to 𝒮 by composition). Both of them have a left adjoint (see [4]). The left adjoint of the forgetful functor from ℳ to 𝒮 is known to be the free adjunction 𝑆¹=𝑆⊔{1} of a unit to a semigroup 𝑆 in order to obtain a monoid in a natural way (the symbol “⊔” denotes the set-theoretical disjoint sum). The unit of this adjunction, 𝑖_𝒮,𝑆∶𝑥∈𝑆→𝑥∈𝑆¹, which is an homomorphism of semigroups, is obviously one-to-one.

The forgetful functor from 𝒢→ℳ has both a left and a right adjoint. Its right adjoint is given, at the object level, as a class mapping that associates a monoid to its group of invertible elements. Its left adjoint, more involved, is described below as group completion.

2.2. Group Completion

The left adjoint of the forgetful functor from groups to monoids may be described as the (unique) solution of the following universal problem. Let 𝑀 be a monoid. Then there exists a unique group 𝓖(𝑀), called the group completion or universal enveloping group or Grothendieck group of 𝑀 (see [6] and references therein, and also [7]), and a unique homomorphism of monoids 𝑖_ℳ,𝑀∶𝑀→𝓖(𝑀) such that for every group 𝐺 and every homomorphism of monoids 𝑓∶𝑀→𝐺, there is a unique homomorphism of groups 𝑓∶𝓖(𝑀)→𝐺 such that the following diagram commutes (in the category of monoids): (1) It is not difficult to check that 𝓖(𝑀) is given either as 𝐹(𝑀)/⟨𝐼_𝑀⟩, where 𝐼_𝑀 is the subset {𝑚𝑛(𝑚∗𝑛)⁻¹∶𝑚,𝑛∈𝑀} (where “∗” is the monoid multiplication of 𝑀, and where 𝐹(𝑋) denotes the free group of 𝑋, see Section  2.1 and if 𝐺 is a group and 𝐴 is any subset of 𝐺, then ⟨𝐴⟩ is the normal subgroup of 𝐺 generated by 𝐴), see [7], or as the quotient monoid (𝑀⊔𝑀⁻¹)^∗/≡_𝑅 where 𝑅={(𝑚𝑚⁻¹,𝜖)∶𝑚∈𝑀}∪{(𝑚⁻¹𝑚,𝜖)∶𝑚∈𝑀} (here the star “∗” stands for the free monoid functor, see also Section  2.1, and 𝜖 is the empty word) where 𝑀⁻¹ is the set of (formal) symbols {𝑚⁻¹∶𝑚∈𝑀} equipotent to 𝑀.

2.3. Free Partially Commutative Structures

Other universal problems, which will play an important role in what follows, are the free partially commutative structures. These structures have been introduced in [8] (see also [9]). A good review of these objects is [10]. Since such constructions may be performed in any of the categories of semigroups, monoids, and groups, they are presented here in a generic way on a category 𝒞∈{𝒮,ℳ,𝒢} so that all statements make sense in any of these categories.

Let 𝑋 be a set and let 𝜃⊆𝑋×𝑋 be a symmetric (i.e., for every 𝑥,𝑦∈𝑋, (𝑥,𝑦)∈𝜃 implies (𝑦,𝑥)∈𝜃) and irreflexive relation on 𝑋 (i.e., for each 𝑥∈𝑋, (𝑥,𝑥)∉𝜃). Let 𝐶 be an object in 𝒞, and 𝑓∶𝑋→𝐶 be a set-theoretical mapping. This function is said to respect the commutations whenever (𝑥,𝑦)∈𝜃 then 𝑓(𝑥)𝑓(𝑦)=𝑓(𝑦)𝑓(𝑥), for every 𝑥,𝑦∈𝑋. A pair (𝑋,𝜃) is called a commutation alphabet.

It can be shown that there exists a unique object 𝒞(𝑋,𝜃) of 𝒞 and a unique mapping 𝑗_𝒞,𝑋∶𝑋→𝐶(𝑋,𝜃) that respects the commutations such that for every object 𝐶 of 𝒞 and every mapping 𝑓∶𝑋→𝐶 that respects the commutations, there is a unique arrow (in 𝒞) 𝑓^𝒞∶𝒞(𝑋,𝜃)→𝐶 such that the following diagram commutes in the category of sets: (2) The object 𝒞(𝑋,𝜃) is usually called the free partially commutative semigroup (resp., monoid, group) on 𝑋 (or on (𝑋,𝜃) to be more precise) depending on 𝒞, and may be constructed as follows: 𝒮(𝑋,𝜃)=𝑋⁺/≡_𝜃 and ℳ(𝑋,𝜃)=𝑋^∗/≡_𝜃 where ≡_𝜃 is the congruence on 𝑋⁺ or 𝑋^∗generated by (𝑥𝑦,𝑦𝑥) whenever (𝑥,𝑦)∈𝜃 for all 𝑥,𝑦∈𝑋 (the least congruence on 𝑋⁺ or 𝑋^∗ containing the relation (𝑥𝑦,𝑦𝑥) whenever (𝑥,𝑦)∈𝜃 for all 𝑥,𝑦∈𝑋, see [11]), and 𝒢(𝑋,𝜃)=𝐹(𝑋,𝜃)/⟨{𝑥𝑦𝑥⁻¹𝑦⁻¹∶(𝑥,𝑦)∈𝜃}⟩.

We may note that 𝒞(𝑋,∅) is nothing else than the usual free (noncommutative) object in the category 𝒞, while 𝒞(𝑋,(𝑋×𝑋)⧵Δ_𝑋), where Δ_𝑋 is the equality relation on 𝑋, is the free commutative object in 𝒞 (in particular, 𝒮(𝑋,(𝑋×𝑋)⧵Δ_𝑋)=𝑋^⊕ is the free commutative semigroup).

We may clarify the relations between the free partially commutative structures. Using universal properties, it is not difficult to check that ℳ(𝑋,𝜃) is isomorphic to 𝒮(𝑋,𝜃)¹ (actually ℳ(𝑋,𝜃)=𝒮(𝑋,𝜃)∪{𝜖}, where 𝜖 is the empty word) in such a way that 𝒮(𝑋,𝜃) embeds in ℳ(𝑋,𝜃) as a subsemigroup.

Lemma 1. The monoid ℳ(𝑋,𝜃) is isomorphic to the free adjunction 𝒮(𝑋,𝜃)¹ of an identity to the semigroup 𝒮(𝑋,𝜃).

Proof. To prove this lemma it is sufficient to check that ℳ(𝑋,𝜃) is a solution of the universal problem of adjunction of a unit to 𝒮(𝑋,𝜃). According to the universal problem of the free partially commutative semigroup 𝒮(𝑋,𝜃), there is a unique homomorphism of semigroups 𝐼∶𝒮(𝑋,𝜃)→ℳ(𝑋,𝜃) such that the following diagram is commutative: (3) Now, let 𝑀 be a monoid and 𝑓∶𝒮(𝑋,𝜃)→𝑀 be a semigroup homomorphism. Therefore there exists 𝑓₀∶𝑋→𝑀 that respects the commutations and such that 𝑓^𝒮₀∘𝑗_𝒮,𝑋=𝑓. According to the universal problem attached to ℳ(𝑋,𝜃), there is a unique homomorphism of monoids 𝑓^ℳ₀∶ℳ(𝑋,𝜃)→𝑓₀ such that 𝑓^ℳ₀∘𝑗_ℳ,𝑋=𝑓₀. Therefore, 𝑓^ℳ₀∘𝐼∘𝑗_𝒮,𝑋=𝑓⁰, but then 𝑓^ℳ₀∘𝐼=𝑓^𝒮₀=𝑓. The relations between all the arrows are summarized in the following commutative diagram: (4)

There is also an important relation between 𝒢(𝑋,𝜃) and ℳ(𝑋,𝜃) given in the following lemma.

Lemma 2. Let (𝑋,𝜃) be a commutation alphabet. Then, 𝒢(𝑋,𝜃) is (isomorphic to) the universal enveloping group 𝓖(ℳ(𝑋,𝜃)) of ℳ(𝑋,𝜃).

Proof. The set-theoretical mapping 𝑗_𝒢,𝑋∶𝑋→𝒢(𝑋,𝜃) respects the commutations, therefore according to the universal problem of the free partially commutative monoid over (𝑋,𝜃) there is a unique homomorphism of monoids 𝑗^ℳ_𝒢,𝑋 that makes commutes the following diagram: (5) Now, let 𝐺 be any group, and 𝑓∶ℳ(𝑋,𝜃)→𝐺 be an homomorphism of monoids. Then, according to the universal problem of the free partially commutative monoid, there is a unique set-theoretical mapping 𝑓₀∶𝑋→𝐺 that respects the commutations and 𝑓∘𝑗_ℳ,𝑋=𝑓₀. Now according to universal problem of 𝒢(𝑋,𝜃), 𝑓₀ is uniquely extended as a group homomorphism 𝑓^𝒢₀∶𝒢(𝑋,𝜃)→𝐺 such that 𝑓^𝒢₀∘𝑗_𝒢,𝑋=𝑓₀. Therefore, 𝑓^𝒢₀∘𝑗^ℳ_𝒢,𝑋∘𝑗_ℳ,𝑋=𝑓^𝒢₀∘𝑗_𝒢,𝑋=𝑓₀=𝑓∘𝑗_ℳ,𝑋 so that 𝑓^𝒢₀∘𝑗^ℳ_𝒢,𝑋=𝑓 (by uniqueness of a solution of a universal problem). Therefore (𝒢(𝑋,𝜃),𝑓^𝒢₀) is a solution of the universal problem of the group completion 𝓖(ℳ(𝑋,𝜃)) of ℳ(𝑋,𝜃). The relations between all the arrows are summarized in the following commutative diagram: (6)

Actually a result from [12, page 66] (see also [10]) states that the natural mapping 𝑗^ℳ_𝒢,𝑋 of the proof of Lemma 2 is one-to-one so that ℳ(𝑋,𝜃) may be identified with a submonoid of its Grothendieck completion 𝒢(𝑋,𝜃).

Definition 3. Let 𝑋 be any set. For every 𝑥∈𝑋 and every 𝑤∈𝑋^∗, let us define |𝑤|_𝑥 as the number of occurrences of the letter 𝑥 in the word 𝑤. More precisely, if 𝜖 is the empty word, then |𝜖|_𝑥=0, |𝑦|_𝑥=0 if 𝑦≠𝑥, |𝑦|_𝑥=1 if 𝑦=𝑥 for all 𝑦∈𝑋, and if the length of 𝑤∈𝑋^∗ is >1, then 𝑤=𝑦𝑤^′ for some letter 𝑦∈𝑋, and 𝑤^′∈𝑋⁺, then |𝑤|_𝑥=|𝑦|_𝑥+|𝑤^′|_𝑥. Let ≡ be a congruence on 𝑋⁺ or 𝑋^∗. It is said to be multihomogeneous if for every 𝑤,𝑤^′ in 𝑋⁺ or 𝑋^∗, such that 𝑤≡𝑤^′, then for every 𝑥∈𝑋, |𝑤|_𝑥=|𝑤^′|_𝑥. Therefore we may define |[𝑤]_≡|_𝑥=|𝑤|_𝑥 for the class [𝑤]_≡ of 𝑤 modulo ≡(it does not depend on the representative of the class modulo ≡).

According to [10], any congruence of the form ≡_𝜃 is a multihomogenous congruence, so that we may define |𝑤|_𝑥 for all 𝑤∈𝒞(𝑋,𝜃) and all 𝑥∈𝑋 (where 𝒞=𝒮 or ℳ). The notion of multihomogeneity is used to check that we may identify the alphabet 𝑋 as a generating set of 𝒞(𝑋,𝜃) using the map 𝑗_𝒞,𝑋, which is shown to be one-to-one, in such a way that we consider that 𝑋⊆𝒞(𝑋,𝜃). Indeed, for semigroup or monoid case, let 𝑥,𝑦∈𝑋 such that their classes modulo ≡_𝜃 be equal. But ≡_𝜃 is a multihomogenous congruence (see [10]). Therefore 𝑥=𝑦. Concerning the group case, let us assume that 𝑥,𝑦∈𝑋 are equivalent modulo the normal subgroup 𝑁_𝜃=⟨{𝑥𝑦𝑥⁻¹𝑦⁻¹∶(𝑥,𝑦)∈𝜃}⟩ so that there is some 𝑤∈𝑁_𝜃 with 𝑥𝑦⁻¹=𝑤. Because the group is free, it means that 𝑥=𝑦 (no nontrivial relations between the generators). In the sequel, we will treat 𝑋 as a subset of 𝒞(𝑋,𝜃).

More generally, let (𝑋,𝜃) be a commutation alphabet and let 𝑌⊆𝑋. We define 𝜃_𝑌=𝜃∩(𝑌×𝑌). It is possible to embed 𝒞(𝑌,𝜃_𝑌) into 𝒞(𝑋,𝜃) as illustrated in the following lemma.

Lemma 4. Under the previous assumptions, there is an arrow 𝐽∶𝒞(𝑌,𝜃_𝑌)→𝒞(𝑋,𝜃) in the category 𝒞 which is one-to-one.

Proof. Let incl∶𝑌→𝑋 be the canonical inclusion. Define 𝐽∶𝒞(𝑌,𝜃_𝑌)→𝒞(𝑋,𝜃) as the unique arrow (in 𝒞) such that the following diagram commutes: (7) Therefore, 𝐽∘𝑗_𝒞,𝑌=𝑗_𝒞,𝑋∘incl.
Let 𝑤₀∈𝒞(𝑌,𝜃_𝑌). Let us define 𝜋_𝑤0∶𝑋→𝑌 such that 𝜋_𝑤0(𝑦)=𝑦 for every 𝑦∈𝑌⊆𝑋, and 𝜋_𝑤0(𝑥)=𝑤₀ for 𝑥∈𝑋⧵𝑌. We note that 𝜋_𝑤0∘incl=𝑖𝑑_𝑌. Then we may consider Π_𝑤0∶𝒞(𝑋,𝜃)→𝒞(𝑌,𝜃_𝑌) as the unique arrow (in 𝒞) that makes commutes the following diagram: (8) Therefore Π_𝑤0∘𝑗_𝒞,𝑋=𝑗_𝒞,𝑌∘𝜋_𝑤0. Now, Π_𝑤0∘𝐽∘𝑗_𝒞,𝑌=Π_𝑤0∘𝑗_𝒞,𝑋∘incl=𝑗_𝒞,𝑌∘𝜋_𝑤0∘incl=𝑗_𝒞,𝑌∘𝑖𝑑_𝑌=𝑗_𝒞,𝑌=𝑖𝑑_{𝒞(𝑌,𝜃𝑌)}∘𝑗_𝒞,𝑌, so that (by uniqueness) Π_𝑤0∘𝐽=𝑖𝑑_{𝒞(𝑌,𝜃𝑌)}, and then 𝐽 is one-to-one (and Π_𝑤0 is onto).

According to Lemma 4 we identify 𝒞(𝑌,𝜃_𝑌) as a subsemigroup, submonoid or subgroup (depending on the choice of 𝒞) of 𝒞(𝑋,𝜃). In such situations we may use the following characterization.

Lemma 5. Let (𝑋,𝜃) be a commutation alphabet, and let 𝑌⊆𝑋 be any subset. Let 𝑤∈𝒮(𝑋,𝜃). The following statements are equivalent.(1)𝑤∈𝒮(𝑌,𝜃_𝑌). (2)For all 𝑥∈𝑋, |𝑤|_𝑥≠0 implies that 𝑥∈𝑌.

Proof. Let 𝑤∈𝒮(𝑋,𝜃). If 𝑤∈𝒮(𝑌,𝜃_𝑌), then for all 𝜔∈𝑋⁺ such that 𝜔∈𝑤, 𝜔∈𝑌⁺ (since 𝒮(𝑌,𝜃_𝑌)≅𝑌⁺/≡_𝜃). Because ≡_𝜃 is a multihomogeneous congruence, |𝜔|_𝑥=|𝑤|_𝑥 for all 𝜔∈𝑤 and 𝑥∈𝑋. Then the point (2). is obtained. Now, let 𝑤∈𝒮(𝑋,𝜃) such that for all 𝑥∈𝑋, |𝑤|_𝑥≠0 implies that 𝑥∈𝑌. Then, for all 𝜔∈𝑤 (𝜔∈𝑋⁺), |𝜔|_𝑥=0 for all 𝑥∉𝑌 which means that 𝜔∈𝑌⁺, and therefore 𝑤∈𝒮(𝑌,𝜃_𝑌) so that (1) is obtained.

3. Basic Notions on Rewriting Systems

3.1. Abstract Rewriting Systems

In this short section, as in the following, we adopt several notations and definitions from [13] that we summarize here.

Let 𝐸 be a set, and ⇒⊆𝐸×𝐸 be any binary relation, called a (one-step) reduction relation, and (𝐸,⇒) is called an abstract rewriting system. We denote by “𝑥⇒𝑦” the membership “(𝑥,𝑦)∈⇒”, and “𝑥⇏𝑦” stands for “(𝑥,𝑦)∉⇒”. Let 𝑅^∗ be the reflexive transitive closure of a binary relation 𝑅. We use 𝑥⇐𝑦 or 𝑥⇐^∗𝑦 to mean that 𝑦⇒𝑥 or 𝑦⇒^∗𝑥. An element 𝑥∈𝐸 is said to be reducible if there exists 𝑦∈𝐸 such that 𝑥⇒𝑦. 𝑥 is irreducible if it is not reducible, or, in other terms, if 𝑥 is ⇒-minimal: there is no 𝑦∈𝐸 such that 𝑥⇒𝑦. A normal form of 𝑥 is an irreducible element 𝑦∈𝐸 such that 𝑥⇒^∗𝑦. If it exists and is unique (see below), the normal form of 𝑥 is denoted by 𝒩(𝑥). The set of all normal forms, or equivalently, of all irreducible elements is denoted by Irr(𝐸,⇒) or Irr(𝐸) when this causes no ambiguity. Note that two distinct normal forms 𝑥,𝑦 are ⇒-incomparable, that is 𝑥⇏𝑦 and 𝑦⇏𝑥. A reduction relation ⇒ is said to be terminating or Noetherian if there is no infinite ⇒-descending chain (𝑥_𝑛)_𝑛∈ℕ of elements of 𝐸 such that 𝑥_𝑛⇒𝑥_𝑛+1 for every 𝑛≥0. In particular, if ⇒ is terminating, then it is irreflexive (otherwise 𝑥_𝑛=𝑥 for some 𝑥∈𝐸 such that 𝑥⇒𝑥∈𝑅 would be an infinite ⇒-descending chain), that is the reason why we freely make use terminology from order relations (such as minimal, Noetherian, descending chain, etc.). We also say that the abstract rewriting system (𝐸,⇒) is terminating or Noetherian whenever ⇒ is so. Two elements 𝑥,𝑦∈𝐸 are said to be joinable if there is some 𝑧∈𝐸 such that 𝑥⇒^∗𝑧⇐^∗𝑦, and ⇒ (and also (𝐸,⇒)) is said to be confluent if for every 𝑥,𝑦₁,𝑦₂∈𝐸 such that 𝑦⇐₁^∗𝑥⇒^∗𝑦₂, then 𝑦₁,𝑦₂ are joinable. A reduction relation ⇒, and an abstract rewriting system (𝐸,⇒), are said to be convergent if it they are both confluent and terminating. Such reduction relations are interesting because in this case any element of 𝐸 has one, and only one, normal form, and if we denote by ⇔^∗ the reflexive transitive symmetric closure of ⇒ (that is the least equivalence relation on 𝐸 containing ⇒), then 𝑥⇔^∗𝑦 if and only if, 𝒩(𝑥)=𝒩(𝑦), therefore 𝒩∶𝐸→Irr(𝐸) satisfies 𝒩(𝒩(𝑥))=𝒩(𝑥) and so is onto and moreover, the function 𝒩∶𝐸/⇔^∗→Irr(𝐸) which maps the class of 𝑥 modulo ⇔^∗ to 𝒩(𝑥) is well defined, onto and one-to-one.

3.2. Semigroup Rewriting Systems

Now, let us assume that 𝐸 is actually a semigroup 𝑆. Let 𝑅⊆𝑆×𝑆 be any binary relation. We define the following relation ⇒_𝑅⊆𝑆×𝑆 by 𝑥⇒_𝑅𝑦 if and only if, there are 𝑢,𝑣∈𝑆¹ and (𝑎,𝑏)∈𝑅 such that 𝑥=𝑢𝑎𝑣 and 𝑦=𝑢𝑏𝑣. A relation ⇒_𝑅 is called the (one-step) reduction rule associated with 𝑅. A relation 𝑅⊆𝑆×𝑆 is said to be two-sided compatible if (𝑥,𝑦)∈𝑅 (𝑥,𝑦∈𝑆) implies (𝑢𝑥𝑣,𝑢𝑦𝑣)∈𝑅. Now, the intersection of the family of all two-sided compatible relations containing a given 𝑅⊆𝑆×𝑆 (this family is nonvoid since it contains the universal relation 𝑆×𝑆) also is a two-sided compatible relation, and so we obtain the least two-sided compatible relation that contains 𝑅. It is called the two-sided compatible relation generated by 𝑅, and it can be shown that this is precisely ⇒_𝑅. Now, given 𝑅⊆𝑆×𝑆, (𝑆,⇒_𝑅) is called a (semigroup) rewriting system; definitions and properties of an abstract rewriting system may be applied to such a rewriting system. When 𝑆 is the free monoid 𝑋^∗, then this kind of rewriting systems is known as string rewriting systems or semi-Thue systems (see [14]). We note that the reflexive transitive symmetric closure ⇔^∗_𝑅 of ⇒_𝑅 is actually a semigroup congruence, because ⇒_𝑅 is two-sided compatible. The quotient semigroup 𝑆/⇔^∗_𝑅 is called the Thue semigroup associated with the semigroup rewriting system (𝑆,⇒_𝑅).

In what follows are considered (particular) string rewriting systems on free partially commutative semigroups. As explained in the end of Section 1, this paper is not a contribution to the theory of string rewriting systems and even not to the well-established theory of trace rewriting systems (that is rewriting systems on free partially commutative monoids). The reader should refer to [15] (in particular to chapters 4 and 5) for more information on this domain. In this work, we consider free partially commutative structures essentially to enlarge the original theory of Brylawski (see also Section  4.3). We recall that our goal is to provide a unified treatment of some similar phenomena of decompositions using the theory of string rewriting systems but for non-specialists.

4. The Tutte-Grothendieck Group of an Alphabetic Rewriting System

4.1. A Free Partially Commutative Structure on Normal Forms

Definition 6. Let (𝑋,𝜃) be a commutation alphabet, and 𝑅⊆𝑋×𝒮(𝑋,𝜃). Then (𝒮(𝑋,𝜃),⇒_𝑅) is called an alphabetic semigroup rewriting system.

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 𝑋 is the set of all (isomorphism classes of) finite multigraphs, or a free (commutative) semigroup or monoid.

Convention. From now on in this current Section  4.1, and only for this subsection, we assume that (𝒮(𝑋,𝜃),⇒_𝑅) is a convergent alphabetic semigroup rewriting system. (In Section  4.2 convergence is not assumed anymore.)

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 12) of this subsection is that the set of all normal forms of a convergent alphabetic semigroup rewriting system is actually the free partially commutative semigroup generated by the irreducible letters.

Lemma 7. Let 𝑤,𝑤^′∈Irr(𝒮(𝑋,𝜃),⇒_𝑅). Then, 𝑤𝑤^′∈Irr(𝒮(𝑋,𝜃),⇒_𝑅). As a result, Irr(𝒮(𝑋,𝜃),⇒_𝑅) is a subsemigroup of 𝒮(𝑋,𝜃).

Proof. Let us assume that 𝑤𝑤^′∉Irr(𝒮(𝑋,𝜃),⇒_𝑅). Therefore there are 𝑥∈𝑋, 𝑤^′′,𝑤^′′′∈𝒮(𝑋,𝜃), 𝑢,𝑣∈ℳ(𝑋,𝜃) such that (𝑥,𝑤^′′′)∈𝑅, 𝑤𝑤^′=𝑢𝑥𝑣 and 𝑤^′′=𝑢𝑤^′′′𝑣 (so that 𝑤𝑤^′⇒_𝑅𝑤^′′). Because ≡_𝜃 is multihomogeneous, either 𝑤=𝑢^′𝑥𝑣^′ or 𝑤^′=𝑢^′𝑥𝑣^′ for some 𝑢^′,𝑣^′∈ℳ(𝑋,𝜃). But in this case, either 𝑤 or 𝑤^′ is reducible, which is a contradiction. As a result, Irr(𝒮(𝑋,𝜃),⇒_𝑅)⊆𝒮(𝑋,𝜃) is closed under the operation of 𝒮(𝑋,𝜃) so that Irr(𝒮(𝑋,𝜃),⇒_𝑅) is a subsemigroup of 𝒮(𝑋,𝜃).

Corollary 8. The map 𝒩∶𝒮(𝑋,𝜃)→Irr(𝒮(𝑋,𝜃),⇒_𝑅) is a surjective homomorphism of semigroups.

Proof. Let 𝑤,𝑤^′∈𝒮(𝑋,𝜃). According to Lemma 7, 𝒩(𝑤)𝒩(𝑤^′)∈Irr(𝒮(𝑋,𝜃),⇒_𝑅). Therefore, 𝒩(𝒩(𝑤)𝒩(𝑤^′))=𝒩(𝑤)𝒩(𝑤^′). Since ⇔^∗ is a congruence of 𝒮(𝑋,𝜃), 𝑤𝑤⇔^′∗𝒩(𝑤)𝒩(𝑤^′) in such a way that 𝒩(𝑤𝑤^′)=𝒩(𝒩(𝑤)𝒩(𝑤^′))=𝒩(𝑤)𝒩(𝑤^′) and 𝒩 is an homomorphism of semigroups. It is obviously onto.

Corollary 9. The semigroups Irr(𝒮(𝑋,𝜃),⇒_𝑅) and 𝒮(𝑋,𝜃)/⇔^∗ are isomorphic.

Proof. As introduced in Section  3.1, let 𝒩∶𝒮(𝑋,𝜃)/⇔^∗→Irr(𝒮(𝑋,𝜃),⇒_𝑅) be the function that maps the class of 𝑤∈𝒮(𝑋,𝜃) modulo ⇔^∗ to the normal form 𝒩(𝑤). It is a one-to-one and onto set-theoretical mapping. But according to Corollary 8, 𝒩 is a semigroup homomorphism, in such a way that 𝒩 also is.

The fact that the rewriting system is alphabetic (Definition 6) actually implies that the (isomorphic) semigroups Irr(𝒮(𝑋,𝜃),⇒_𝑅) and 𝒮(𝑋,𝜃)/⇔^∗ are actually free partially commutative. The objective is now to prove this statement. In order to do that, we exhibit the commutation alphabet that generates them. Let Irr(𝑋)=Irr(𝒮(𝑋,𝜃),⇒_𝑅)∩𝑋 (recall from Section  2.3 that 𝑋 is considered as a subset of 𝒮(𝑋,𝜃)). It is clear that Irr(𝑋)={𝑥∈𝑋∶∄𝑤∈𝑆(𝑋,𝜃),(𝑥,𝑤)∈𝑅}. Indeed, for every 𝑥∈𝑋, 𝑤∈𝒮(𝑋,𝜃), 𝑥⇒_𝑅𝑤 if and only if, there are 𝑢,𝑣∈ℳ(𝑋,𝜃), 𝑥₁∈𝑋, 𝑤₁∈𝒮(𝑋,𝜃) such that 𝑥=𝑢𝑥₁𝑣 and 𝑤=𝑢𝑤₁𝑣. Since ≡_𝜃 is a multihomogenous congruence (see Section  2.3), 𝑢=𝑣 is the empty word, and 𝑥=𝑥₁, 𝑤=𝑤₁. Therefore 𝑥⇒_𝑅𝑤 if and only if, (𝑥,𝑤)∈𝑅.

This characterization of Irr(𝑋) is used in the following lemma.

Lemma 10. If 𝑋≠∅, then Irr(𝑋)≠∅.

Proof. Let us assume that 𝑋≠∅ and Irr(𝑋)=∅. Let 𝑥∈𝑋. Since 𝑥∉Irr(𝑋), there is some 𝑤∈𝒮(𝑋,𝜃) such that (𝑥,𝑤)∈𝑅. Because 𝑤∈𝒮(𝑋,𝜃), and 𝑋 generates 𝒮(𝑋,𝜃), it can be written as 𝑥₁𝑢 for some 𝑥₁∈𝑋, and 𝑢∈ℳ(𝑋,𝜃). Because 𝑥₁∉Irr(𝑋), there is 𝑣₁∈𝒮(𝑋,𝜃) such that (𝑥₁,𝑣₁)∈𝑅. Then, 𝑤⇒_𝑅𝑣₁𝑢. Replacing 𝑤 by 𝑣₁𝑢, we may construct an infinite descending chain 𝑥⇒_𝑅𝑤⇒_𝑅𝑣₁𝑢⇒_𝑅⋯, which is impossible since ⇒_𝑅 is assumed to be convergent, and therefore terminating. So Irr(𝑋)≠∅.

Remark 11. Forthcoming Proposition 12, Lemma 14 and Corollary 15 and Theorem 16 are obviously valid when 𝑋=∅.

The following lemma reveals the structure of free partially commutative semigroup of Irr(𝒮(𝑋,𝜃),⇒_𝑅), and therefore also of 𝒮(𝑋,𝜃)/⇔^∗ according to Corollary 9.

Proposition 12. The semigroup Irr(𝒮(𝑋,𝜃),⇒_𝑅) is equal to the free partially commutative semigroup 𝒮(Irr(𝑋),𝜃_Irr(𝑋)) where 𝜃_Irr(𝑋)=𝜃∩(Irr(𝑋)×Irr(𝑋)) (see Lemma 4).

Proof. Let 𝑤∈Irr(𝒮(𝑋,𝜃),⇒_𝑅). Let us assume that 𝑤∉𝒮(Irr(𝑋),𝜃_Irr(𝑋)). According to Lemma 5, there exists 𝑥∈𝑋⧵Irr(𝑋) such that for all 𝜔∈𝑋⁺, 𝜔∈𝑤 (𝑤 is seen as a congruence class), |𝜔|_𝑥≠0. Therefore 𝜔=𝑢𝑥𝑣 for some 𝑢,𝑣∈𝑋^∗ and 𝑤=𝜋_𝜃(𝑢)𝑥𝜋_𝜃(𝑣) (where 𝜋_𝜃∶𝑋^∗→ℳ(𝑋,𝜃) is the canonical epimorphism and where we recall that 𝑋 is seen as a subset of 𝒮(𝑋,𝜃), 𝑋^∗=𝑋⁺⊔{𝜖}, and ℳ(𝑋,𝜃)=𝒮(𝑋,𝜃)⊔{𝜖}). But 𝑥∉Irr(𝑋), then there exists 𝑤^′∈𝒮(𝑋,𝜃) such that (𝑥,𝑤^′)∈𝑅, and therefore 𝑤⇒_𝑅𝜋_𝜃(𝑢)𝑤^′𝜋_𝜃(𝑣) which contradicts the fact that 𝑤∈Irr(𝒮(𝑋,𝜃),⇒_𝑅). Let 𝑤∈𝒮(𝑋,𝜃) such that 𝑤∈𝒮(Irr(𝑋),𝜃_Irr(𝑋)). Let us assume that 𝑤∉Irr(𝒮(𝑋,𝜃),⇒_𝑅). Therefore 𝑤=𝑢𝑥𝑣 for some 𝑢,𝑣∈ℳ(𝑋,𝜃), 𝑥∈𝑋 such that there is 𝑤^′∈𝒮(𝑋,𝜃) with (𝑥,𝑤^′)∈𝑅. Therefore 𝑥∉Irr(𝑋). It is then clear that for every 𝜔∈𝑋⁺ such that 𝜔∈𝑤, |𝜔|_𝑥>0. But according to Lemma 5, this is impossible because 𝑥∉Irr(𝑋) and 𝑤∈𝒮(Irr(𝑋),𝜃_Irr(𝑋)). We have proved that Irr(𝒮(𝑋,𝜃),⇒_𝑅) and 𝒮(Irr(𝑋),𝜃_Irr(𝑋)) are equal as sets. But since they are both subsemigroups of 𝒮(𝑋,𝜃), then they are equal as semigroups.

4.2. The Tutte-Grothendieck Group of an Alphabetic Rewriting System

Definition 13. Let (𝑋,𝜃) be a commutation alphabet, and let ⇒_𝑅 be an alphabetic rewriting system. Let 𝑆 be any semigroup, and let 𝑓∶𝑋→𝑆 that respects the commutations. Let 𝑓^𝒮∶𝒮(𝑋,𝜃)→𝑆 be the unique homomorphism of semigroups such that the following diagram commutes (see Section 2.3): (9) Then 𝑓 is said to be an 𝑅-invariant if for every 𝑥∈𝑋 and 𝑤∈𝒮(𝑋,𝜃) such that (𝑥,𝑤)∈𝑅, then 𝑓(𝑥)=𝑓^𝒮(𝑤).

Informally speaking, according to Definition 13, a function 𝑓 that respects the commutations is an 𝑅-invariant if its canonical semigroup extension 𝑓^𝒮 is constant for all reductions (𝑥,𝑤)∈𝑅.

Let us assume that (𝑋,𝜃) is a commutation alphabet, and let ⇒_𝑆 be an alphabetic rewriting system on 𝒮(𝑋,𝜃) (not necessarily convergent). The fact that the rewriting system is alphabetic implies in an essential way the following results.

Lemma 14. Let 𝑆 be a semigroup, and let 𝑓∶𝑋→𝑆 be a function that respects the commutations. Let 𝑓^𝒮 be its canonical semigroup extension from 𝒮(𝑋,𝜃) to 𝑆. If 𝑓 is a 𝑅-invariant, then for every 𝑤,𝑤^′∈𝒮(𝑋,𝜃) such that 𝑤⇒_𝑅𝑤^′, one has 𝑓^𝒮(𝑤)=𝑓^𝒮(𝑤^′).

Proof. Since we will deal with the empty word, one needs to recall the following. According to Lemma 1, ℳ(𝑋,𝜃)=𝒮(𝑋,𝜃)∪{𝜖}, where 𝜖 is the empty word. Let us define 𝑓^𝒮₁∶ℳ(𝑋,𝜃)→𝑆¹ the canonical extension of 𝑓^𝒮 as a monoid homomorphism. That is, whenever 𝑤∈𝒮(𝑋,𝜃), 𝑓^𝒮₁(𝑤)=𝑓^𝒮(𝑤), and 𝑓^𝒮₁(𝜖)=1. Let 𝑤,𝑤^′∈𝒮(𝑋,𝜃) such that 𝑤⇒_𝑅𝑤^′. Then there exist 𝑥∈𝑋, 𝑤^′′∈𝒮(𝑋,𝜃), 𝑢,𝑣∈ℳ(𝑋,𝜃) such that (𝑥,𝑤^′′)∈𝑅, 𝑤=𝑢𝑥𝑣 and 𝑤^′=𝑢𝑤^′′𝑣. Because 𝑓 is 𝑅-invariant, 𝑓(𝑥)=𝑓^𝒮(𝑤^′), and then we have 𝑓^𝒮(𝑤)=𝑓^𝒮(𝑢𝑥𝑣)=𝑓^𝒮₁(𝑢𝑥𝑣)=𝑓^𝒮₁(𝑢)𝑓^𝒮₁(𝑥)𝑓^𝒮₁(𝑣)=𝑓^𝒮₁(𝑢)𝑓^𝒮(𝑥)𝑓^𝒮₁(𝑣)=𝑓^𝒮₁(𝑢)𝑓(𝑥)𝑓^𝒮₁𝑓(𝑣)=_𝒮1(𝑢)𝑓𝒮(𝑤^′′)𝑓^𝒮₁(𝑣)=𝑓^𝒮₁(𝑢)𝑓𝒮₁(𝑤^′′)𝑓^𝒮₁(𝑣)=𝑓^𝒮₁(𝑢𝑤^′′𝑣)=𝑓^𝒮₁(𝑤^′)=𝑓^𝒮(𝑤^′).

Corollary 15. Let 𝑆 be a semigroup, and let 𝑓∶𝑋→𝑆 be a function that respects the commutations. If 𝑓 is a 𝑅-invariant, then its canonical semigroup extension 𝑓^𝒮 factors through the quotient 𝒮(𝑋,𝜃)/⇔^∗_𝑅.

Proof. Let 𝑤,𝑤^′∈𝒮(𝑋,𝜃) such that 𝑤⇔^∗_𝑅𝑤^′. Then there are 𝑛>0, 𝑤₀,…,𝑤_𝑛∈𝒮(𝑋,𝜃), 𝑤₀=𝑤, 𝑤_𝑛=𝑤^′ such that for every 0≤𝑖<𝑛, 𝑤_𝑖=𝑤_𝑖+1 or 𝑤_𝑖⇔_𝑅𝑤_𝑖+1. Therefore for every 0≤𝑖<𝑛, either 𝑤_𝑖=𝑤_𝑖+1, or 𝑤_𝑖⇒_𝑅𝑤_𝑖+1, or 𝑤_𝑖⇐_𝑅𝑤_𝑖+1. Because 𝑓 is a 𝑅-invariant, according to Lemma 14, for every 0≤𝑖<𝑛, 𝑓^𝒮(𝑤_𝑖)=𝑓^𝒮(𝑤_𝑖+1). Therefore 𝑓^𝒮(𝑤)=𝑓^𝒮(𝑤₀)=⋯=𝑓^𝒮(𝑤_𝑛)=𝑓^𝒮(𝑤^′). Then, there exists a unique semigroup homomorphism 𝑓^𝒮_𝑅∶𝒮(𝑋,𝜃)/⇔^∗_𝑅→𝑆 such that 𝑓^𝒮_𝑅([𝑤]^∗_⇔𝑅)=𝑓^𝒮(𝑤) for every 𝑤∈𝒮(𝑋,𝜃) (where [𝑤]^∗_⇔𝑅 denotes the class of 𝑤 modulo ⇔^∗_𝑅).

We are now in position to establish the main result of this paper.

Theorem 16. Let (𝑋,𝜃) be a commutation alphabet, and let (𝒮(𝑋,𝜃),⇒_𝑅) be an alphabetic rewriting system. There exist a group 𝒯𝒢(𝑋,𝜃,𝑅) and a mapping 𝑡∶𝑋→𝒯𝒢(𝑋,𝜃,𝑅) that respects the commutations which is an 𝑅-invariant such that for every group 𝐺, and every (commutations respecting) 𝑅-invariant mapping 𝑓∶𝑋→𝐺, there is a unique group homomorphism ℎ∶𝒯𝒢(𝑋,𝜃,𝑅)→𝐺 such that the following diagram commutes: (10) Moreover, if ⇒_𝑅 is convergent, then the group 𝒯𝒢(𝑋,𝜃,𝑅) is isomorphic to the free partially commutative group 𝒢(Irr(𝑋),𝜃_Irr(𝑋)) and 𝑡 is the normal form 𝒩∘𝑗_𝒮,𝑋∶𝑋→𝒮(Irr(𝑋),𝜃_Irr(𝑋)) restricted to the alphabet 𝑋 (recall that one has 𝑋⊆𝒮(Irr(𝑋),𝜃_Irr(𝑋))⊆ℳ(Irr(𝑋),𝜃_Irr(𝑋))⊆𝒢(Irr(𝑋),𝜃_Irr(𝑋)) under natural identifications; see Section  2.3).

Proof. Let 𝐺 be a group and let 𝑓∶𝑋→𝐺 be a commutations respecting 𝑅-invariant mapping. According to the universal problem of free partially commutative semigroups, because 𝐺 is also a semigroup, we have the following commutative diagram: (11) According to Corollary 15, we may complete the previous diagram in a natural way (the notations from the proof of Corollary 15 are used) as fellows: (12) Now, we extend in a natural way 𝑓^𝒮_𝑅 as a monoid homomorphism 𝑓^𝒮_𝑅,1∶(𝒮(𝑋,𝜃)/⇔^∗_𝑅)¹→𝐺 (because 𝐺 is also a monoid). Let us denote by 𝑀 the monoid (𝒮(𝑋,𝜃)/⇔^∗_𝑅)¹. We obtain the following diagram: (13) Finally, using the Grothendieck group 𝓖(𝑀) of 𝑀, we complete the previous commutative diagram as follows (where 𝑖=𝑖_{𝒮,𝒮(𝑋,𝜃)/}^∗_⇔𝑅∘[⋅]^∗_⇔𝑅∘𝑗_𝒮,𝑋): (14) Now, as illustrated in the previous diagram, let 𝒯𝒢(𝑋,𝜃,𝑅)=𝓖(𝑀), 𝑡=𝑖_ℳ,𝑀∘𝑖_{𝒮,𝒮(𝑋,𝜃)/}^∗_⇔𝑅∘[⋅]^∗_⇔𝑅∘𝑗_𝒮,𝑋, and ℎ=𝑓^𝒮_𝑅,1. First of all, 𝑡 obviously respects the commutations. Let us consider the canonical extension 𝑡^𝒮∶𝑋→𝒮(𝑋,𝜃) of 𝑡. So we have the following commutative diagram: (15) By uniqueness of the solution of a universal problem, and according to the diagram (14), we have 𝑡^𝒮=𝑖_ℳ,𝑀∘𝑖_{𝒮,𝒮(𝑋,𝜃)/}^∗_⇔𝑅∘[⋅]^∗_⇔𝑅. Now, let 𝑥∈𝑋, 𝑤∈𝒮(𝑋,𝜃) such that (𝑥,𝑤)∈𝑅. Then, [𝑥]^∗_⇔𝑅=[𝑤]^∗_⇔𝑅. Therefore, 𝑡(𝑥)=𝑡^𝒮(𝑥)=𝑡^𝒮(𝑤), so that 𝑡 is 𝑅-invariant. Then the first part of the theorem is proved.
Now, let us assume that (𝒮(𝑋,𝜃),⇒_𝑅) is convergent. Then, by Proposition 12, 𝒮(𝑋,𝜃)/⇔^∗_𝑅 is isomorphic to the free partially commutative semigroup 𝒮(Irr(𝑋),𝜃_Irr(𝑋)). Therefore, 𝑀=(𝒮(𝑋,𝜃)/⇔^∗_𝑅)¹ is isomorphic to the free partially commutative monoid ℳ(𝑋,𝜃) (by Lemma 1). Finally, the Grothendieck group 𝓖(𝑀) is isomorphic to the Grothendieck group 𝓖(ℳ(𝑋,𝜃)) (because 𝓖(⋅) is functorial) so that it is isomorphic to the free partially commutative group 𝒢(𝑋,𝜃) (by Lemma 2). The fact that in this case, 𝑡 is the normal form 𝒩∘𝑗_𝒮,𝑋∶𝑋→𝒮(Irr(𝑋),𝜃_Irr(𝑋)) restricted to the alphabet 𝑋 (where 𝒮(Irr(𝑋),𝜃_Irr(𝑋)) is naturally identified with a subsemigroup of 𝒢(Irr(𝑋),𝜃_Irr(𝑋))) is quite obvious to check.

Definition 17. The group 𝒯𝒢(𝑋,𝜃,𝑅) is called the Tutte-Grothendieck group and 𝑡 the universal Tutte-Grothendieck 𝑅-invariant of the alphabetic rewriting system (𝒮(𝑋,𝜃),⇒_𝑅).

4.3. Some Examples

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).

4.3.1. The Tutte Polynomial

In its famous paper [1], Tutte used the following decomposition of (isomorphism classes of) finite multigraphs (graphs with multiple edges and loops). Let 𝐺 be a multigraph, and 𝑒 be a link (edge which is not a loop nor a bridge) in 𝐺. Let 𝐺−𝑒 be the graph obtained from 𝐺 by erasing 𝑒, and let 𝐺/𝑒 be the graph obtained by contraction of 𝑒 in 𝐺 (𝑒 is removed, and its origin and source are identified). Then 𝐺 is decomposed into (𝐺−𝑒)+𝐺/𝑒 (+ being the free commutative juxtaposition). As explained in [3] in terms of an order relation, a rewriting system may be defined, and the universal invariant attached to this system is the well-known Tutte polynomials (see [1]). In [16] Stanley uses the theory developed by Brylawski to define a similar decomposition on finite ordered sets.

4.3.2. Integral Weyl Algebra

For any set 𝑋, let 𝑋^⊕ be the free commutative semigroup generated by 𝑋 (i.e., 𝑋^⊕=𝒮(𝑋,𝜃), where 𝜃=(𝑋×𝑋)⧵Δ_𝑋 and Δ_𝑋 is the equality relation on 𝑋), written additively. Recall also that the free Abelian group generated by 𝑋, namely 𝒢(𝑋,𝜃), is isomorphic to the group (under point-wise addition) ℤ^(𝑋) of all mappings from 𝑋 to ℤ with a finite support (the support of a function 𝑓∶𝑋→ℤ is the set of all 𝑥∈𝑋 such that 𝑓(𝑥)≠0), see for instance [5]. Let 𝑌={𝑎,𝑏} be a two element set. Let 𝑋=𝑌^∗, and 𝑅={(𝑢𝑎𝑏𝑣,𝑢𝑏𝑎𝑣+𝑢𝑣)∶𝑢,𝑣∈𝑌^∗}⊆𝑋×𝑋^⊕. It is clear that Irr(𝑋)={𝑏^𝑖𝑎^𝑗∶𝑖,𝑗∈ℕ}. Moreover the alphabetic rewriting system (𝑋^⊕,⇒_𝑅) is convergent (it is not difficult to check this property using for instance techniques from [17]). Let 𝜃=(𝑋×𝑋)⧵Δ_𝑋. Then 𝒯𝒢(𝑋,𝜃,𝑅)=𝒢(Irr(𝑋),𝜃_Irr(𝑋))=ℤ^(Irr(𝑋)). Therefore we recover the well-known fact (see [18]) that the integral Weyl algebra 𝐴_ℤ=ℤ⟨𝑎,𝑏⟩/𝐼_[𝑎,𝑏] with two generators (where ℤ⟨𝑎,𝑏⟩ denotes the ring of the free monoid 𝑋=𝑌^∗={𝑎,𝑏}^∗, and where 𝐼_[𝑎,𝑏] is the two-sided ideal of ℤ⟨𝑎,𝑏⟩ generated by 𝑎𝑏−𝑏𝑎−1) is free as an Abelian group with generators Irr(𝑋). The universal Tutte-Grothendieck 𝑅-invariant 𝑡 of (𝑋^⊕,⇒_𝑅) is the normal form of the words in 𝑋=𝑌^∗. For instance, 𝑡(𝑏𝑎𝑏𝑎𝑏)=𝑏²𝑎²+3𝑏²𝑎+𝑏.

Let 𝑐 be a variable (distinct from 𝑎,𝑏) and let 𝑌_𝑐=𝑌∪{𝑐}={𝑎,𝑏,𝑐}. Consider the relation 𝜃={(𝑥,𝑐)∶𝑥∈𝑌}∪{(𝑐,𝑥)∶𝑥∈𝑌}. Finally let 𝑋_𝑐=𝒮(𝑌_𝑐,𝜃). Therefore 𝑐 commutes with all elements of 𝑋_𝑐. Let 𝑅_𝑐={(𝑢𝑎𝑏𝑣,𝑢𝑏𝑎𝑣+𝑢𝑣)∶𝑢,𝑣∈ℳ(𝑌_𝑐,𝜃)}⊆𝑋_𝑐×𝑋^⊕_𝑐. Then we can check that (𝑋^⊕_𝑐,⇒_𝑅𝑐) is a convergent alphabetic rewriting system whose Tutte-Grothendieck group is ℤ^{Irr(𝑋𝑐)} where Irr(𝑋_𝑐)={𝑐^𝑖𝑏^𝑗𝑎^𝑘∶𝑖,𝑗,𝑘∈ℕ} (note that 𝑐^𝑖𝑏^𝑗𝑎^𝑘=𝑐^𝑖1𝑏^𝑗𝑐^𝑖2𝑎^𝑘𝑐^𝑖3 for every nonnegative integers 𝑖₁, 𝑖₂, 𝑖₃, 𝑖=𝑖₁+𝑖₂+𝑖₃, 𝑗 and 𝑘, since 𝑐 commutes with all other elements). This gives us immediately a free ℤ-basis for the central extension ℤ⟨𝑎,𝑏,𝑐⟩/𝐼_{[𝑎,𝑏],𝑐} (where 𝐼_{[𝑎,𝑏],𝑐} is the two-sided ideal of the ring ℤ⟨𝑎,𝑏,𝑐⟩ of the monoid 𝑌^∗_𝑐 generated by 𝑎𝑏−𝑏𝑎−1 and 𝑐𝑥−𝑥𝑐 for every 𝑥∈{𝑎,𝑏}) of the integral Weyl algebra 𝐴_ℤ.

4.3.3. Integral Temperley-Lieb Algebra

Let 𝑛∈ℕ such that 𝑛>2. Let us consider 𝑌={𝑥₁,…,𝑥_𝑛−1}. Let 𝜃={(𝑥_𝑖,𝑥_𝑗)∶|𝑖−𝑗|>1}∪{(𝑥_𝑗,𝑥_𝑖)∶|𝑖−𝑗|>1}. Let 𝑋=𝒮(𝑌,𝜃). Let 𝑅={(𝑢𝑥_𝑖𝑥_𝑗𝑥_𝑖𝑣,𝑢𝑥_𝑖𝑣)∶|𝑖−𝑗|=1,𝑢,𝑣∈ℳ(𝑌,𝜃)}∪{(𝑢𝑥²_𝑖𝑣,𝑢𝑥_𝑖𝑣)∶1≤𝑖≤𝑛−1,𝑢,𝑣∈ℳ(𝑌,𝜃)}⊆𝑋×𝑋^⊕. This alphabetic rewriting system is easily shown to be convergent. The corresponding Tutte-Grothendieck ring is ℤ^(𝑌). We recover the fact that the 𝑛-th Temperley-Lieb ℤ-algebra (with parameter 1) ℤ⟨𝑌⟩/𝐼, where 𝐼 the two-sided ideal generated by the following relations 𝑥_𝑖𝑥_𝑗=𝑥_𝑗𝑥_𝑖 for |𝑖−𝑗|>1, 𝑥_𝑖𝑥_𝑗𝑥_𝑖=𝑥_𝑖 for |𝑖−𝑗|=1, 𝑥²_𝑖=𝑥_𝑖 for 1≤𝑖≤𝑛−1 is free with basis 𝑌.

4.3.4. The Poincaré-Birkhoff-Witt Theorem

Let 𝔤 be a Lie algebra over ℤ which is free as an ℤ-module (see [19]). Let ℬ be a basis of 𝔤 seen as a (free) ℤ-module. Let us assume that ℬ is linearly ordered by ≤. Let 𝑋=ℬ^∗ be the free monoid generated by ℬ. Let 𝑅={(𝑢ℎ𝑔𝑣,𝑢𝑔ℎ𝑣)∶𝑔,ℎ∈ℬ,𝑔<ℎ,𝑢,𝑣∈ℬ^∗}⊆𝑋×𝑋^⊕. It is obvious that (𝑋^⊕,⇒_𝑅) is a convergent alphabetic rewriting system. Moreover, Irr(𝑋)={𝑔₁⋯𝑔_𝑛∶𝑛≥0,𝑔_𝑖∈ℬ for all 0≤𝑖≤𝑛,𝑔_𝑖≤𝑔_𝑖+1forall0≤𝑖<𝑛} and its Tutte-Grotendieck group is ℤ^(Irr(𝑋)), while its universal Tutte-Grothendieck invariant 𝑡 is the reordering of an element of 𝑋 in an increasing order (relative to ≤). We recognize the famous Poincaré-Birkhoff-Witt theorem ([20–22]) since the universal enveloping ℤ-algebra of 𝔤 over ℤ is free with basis Irr(𝑋).

4.3.5. Prefabs

In [23], Bender and Goldman introduced the notion of a prefab for combinatorial purposes (computation of some generating functions). We recall here (a part of) this concept. Let 𝑋 be a set together with a multivalued binary operation ∘ (meaning that 𝑥,𝑦∈𝑋 implies that 𝑥∘𝑦⊆𝑋) subjected to properties given below. For every 𝑥,𝑦∈𝑋, 𝑥∘𝑦 is a finite set. The operation ∘ is extended to the power set 2^𝑋 of 𝑋 by 𝐴∘𝐵={𝑧∈𝑆∶𝑥∈𝑥∘𝑦 for some 𝑥∈𝐴,𝑦∈𝐵}. If 𝑥∈𝑋 and 𝐴⊆𝑋, then we let 𝑥∘𝐴 be equal to {𝑥}∘𝐴=𝐴∘{𝑥}, and 𝑥^𝑖 is defined by induction: 𝑥¹={𝑥}, and 𝑥^𝑖+1=𝑥∘𝑥^𝑖 for every positive integer 𝑖. We say that (𝑋,∘) is a prefab if the composition ∘ on 2^𝑋 is associative, commutative (therefore 2^𝑋 becomes a semigroup), and has an identity (the identity plays also a role in counting arguments in [23]) 𝑖∈𝑆 such that 𝑥∘𝑖={𝑥}=𝑖∘𝑥 for every 𝑥∈𝑋 (then 2^𝑋 is a monoid). An element 𝑝∈𝑋⧵{𝑖} is called a prime if 𝑝∈𝑥∘𝑦 implies 𝑥=𝑖 or 𝑦=𝑖. We say that (𝑋,∘) is a unique factorization prefab if every 𝑥∈𝑋⧵{𝑖} factors uniquely into primes in the sense that 𝑥∈𝑝^𝑖1₁∘⋯∘𝑝^𝑖𝑛_𝑛 for a unique set of 𝑛>0 primes {𝑝_𝑖∶1≤𝑖≤𝑛} and unique positive integers 𝑖₁,…,𝑖_𝑛. We say that (𝑋,∘) is a very unique factorization prefab if 𝑥∈(𝑝^𝑖1₁∘⋯∘𝑝^𝑖𝑚_𝑚)∘(𝑞^𝑗1₁∘⋯∘𝑞^𝑗𝑛_𝑛) where 𝑚>0,  𝑛>0, all the 𝑖’s and all the 𝑗’s are positive integers, all the 𝑝’s are mutually distinct primes, and all the 𝑞’s are mutually distinct primes (but some 𝑞’s may be equal to some 𝑝’s), then there exist unique elements 𝑦∈𝑝^𝑖1₁∘⋯∘𝑝^𝑖𝑚_𝑚 and 𝑧∈𝑞^𝑗1₁∘⋯∘𝑞^𝑗𝑛_𝑛 such that 𝑥∈𝑦∘𝑧. In the original definition of a prefab, there is also a mapping 𝑓∶2^𝑋→ℕ which serves as a weigth function for a combinatorial use but which is not needed here.

Let (𝑌,∘) be a unique and very unique factorization prefab. Let 𝑃 be the set of primes of this prefab. Let 𝑋=𝑌⧵{𝑖}. Let 𝑅={(𝑥,𝑦+𝑧)∶∃𝑦,𝑧∈𝑋,𝑥∈𝑦∘𝑧}⊆𝑋×𝑋^⊕. According to the properties of unique factorization, very unique factorization, associativity, and commutativity of ∘, it is clear that (𝑋^⊕,⇒_𝑅) is a convergent alphabetic rewriting system. We have Irr(𝑋)=𝑃, and the Tutte-Grothendieck group is, as expected, ℤ^(𝑃). It is also immediate that 𝑡(𝑥)=∑^𝑛_𝑗=1𝑖_𝑗𝑝_𝑗 where 𝑝^𝑖1₁∘⋯∘𝑝^𝑖𝑛_𝑛 is the unique prime factorization of 𝑥.

As examples of (unique and very unique factorization) prefabs, one can cite the following two from [23]. Let 𝑋 be any set, and let 𝑤,𝑤^′∈𝑋⁺ be two words. A shuffle of these two words is a word 𝑤^′′=𝑥₁⋯𝑥_𝑛∈𝑋⁺, 𝑥_𝑖∈𝑋 for 1≤𝑖≤𝑛 (where 𝑛 is the sum of lengths of 𝑤 and 𝑤^′) such that there exists {𝐼,𝐽} a partition of {1,…,𝑛} with 𝑤=𝑥_𝑖1⋯𝑥_𝑖𝑘, 𝑖₁<⋯<𝑖_𝑘, 𝑘 is the cardinal of 𝐼, 𝐼={𝑖₁,…,𝑖_𝑘}, and 𝑤^′=𝑥_𝑗1⋯𝑥_𝑗ℓ, 𝑗₁<⋯<𝑗_ℓ, ℓ is the cardinal of 𝐽, 𝐽={𝑗₁,…,𝑗_ℓ} (such constructions appear in the shuffle product of two words; see [24]). Let 𝑤∘𝑤^′ be the set of all shuffles of 𝑤 and 𝑤^′. As an example, 𝑤=𝛼𝛾 and 𝑤^′=𝛽𝛽. Then 𝑤∘𝑤^′={𝛼𝛾𝛽𝛽,𝛼𝛽𝛾𝛽,𝛼𝛽𝛽𝛾,𝛽𝛼𝛾𝛽,𝛽𝛼𝛽𝛾,𝛽𝛽𝛼𝛾}. It is clear that the identity is the empty word (therefore we allow to choose word in 𝑋^∗) while the prime elements are the letters in 𝑋. The prime decomposition of a word is then the set of the letters that form the words. The rewriting system associated to this prefab is the following: (𝑤,𝑤^′+𝑤^′′) where 𝑤^′+𝑤^′′∈(𝑋⁺)^⊕ such that 𝑤∈𝑤^′∘𝑤^′′. To summarize, the set Irr(𝑋⁺) is 𝑋, the Tutte-Grothendieck group is ℤ^(𝑋), and the universal invariant is given by 𝑡(𝑤)=∑_𝑥∈𝑋|𝑤|_𝑥𝑥 (which is sometimes called the commutative image; see [25]).

Let 𝑥_𝑖 be an indeterminate for each 𝑖∈ℕ⧵{0} such that 𝑥_𝑖≠𝑥_𝑗 whenever 𝑖≠𝑗. Let 𝑌={𝑥_𝑖∶𝑖≥1}. Let 𝐷(𝑥_𝑛)={∑_𝑖>1𝑘_𝑖𝑥_𝑖∈𝑌^⊕∶𝑘_𝑖∈ℕ,forall𝑖>1,𝑘_𝑖=0 except a finite number, ∑_𝑖>1𝑘_𝑖𝑖=𝑛}⊆𝑌^⊕. Finally let us define 𝑥_𝑚∘𝑥_𝑛={𝑓+𝑔∈𝑌^⊕∶𝑓∈𝐷(𝑥_𝑚),𝑔∈𝐷(𝑥_𝑛)}. For instance, 𝑥₈∘𝑥₄={6𝑥₂,3𝑥₂+𝑥₄,𝑥₂+2𝑥₄,2𝑥₂+𝑥₈,𝑥₄+𝑥₈}. The identity is 𝑥₁, while the primes are exactly the 𝑥_𝑝 for 𝑝∈ℙ, where ℙ is the set of all prime integers. Attached with these datas, the rewriting system on (𝑌⧵{𝑥₁})^⊕ is given by 𝑅={(𝑥_𝑛,𝑓)∶𝑓∈𝐷(𝑥_𝑛)}. The Tutte-Grothendieck group is ℤ^(ℙ), and the universal invariant is given by 𝑡(𝑥_𝑚)=∑^ℓ_𝑖=1𝑘_𝑖𝑥_𝑝𝑖, where 𝑝^𝑘1₁⋯𝑝^𝑘ℓ_ℓ is the decomposition of 𝑚 into prime numbers.

Appendix

A Short Review of Brylawski’s Theory

In this appendix are briefly presented the main definitions and results of Brylawski’s theory that are extended and clarified in this contribution.

Let 𝑋 be a set, and let 𝐷(𝑋)⊆𝑋^⊕. Let (𝐷(𝑋),≤) be a partially ordered set with 𝐷(𝑋)⊆𝑋^⊕ such that (1)for every 𝑓,𝑔∈𝐷(𝑋), if 𝑓≤𝑔, then |𝑓|<|𝑔| or 𝑓=𝑔 (where |𝑓|=∑_𝑥∈𝑋𝑓(𝑥)), (2)for every 𝑓,𝑔,ℎ∈𝐷(𝑋), 𝑓+𝑔≤ℎ if and only if, there exist ℎ₁,ℎ₂ in 𝐷(𝑆) such that ℎ₁+ℎ₂=ℎ, 𝑓≤ℎ₁, and 𝑔≤ℎ₂.

A partial ordered set of this kind is called a decomposition of 𝑆, and we say that 𝑓decomposes into 𝑔 when 𝑓≤𝑔. An element 𝑥 of 𝑋∩𝐷(𝑋) is said to be irreducible if 𝑥 is maximal with respect to ≤. According to axiom (1), the elements of 𝑋 that belong to 𝐷(𝑋) are minimal with respect to ≤, therefore the irreducible elements are the incomparable elements. Let us denote by Irr(𝑋) their totality. A decomposition 𝐷(𝑋) is said to be finite when for every 𝑥∈𝑋, there exists 𝑓∈𝐷(𝑋)∩Irr(𝑋)^⊕⊆𝑋^⊕ such that 𝑥≤𝑓 (we say that 𝑥  fully decomposed into 𝑓); in particular 𝑋⊆𝐷(𝑋). A decomposition 𝐷(𝑋) is said to be refinable if 𝑓≤𝑔 and 𝑓≤ℎ imply that there is 𝑘∈𝐷(𝑋) such that 𝑔≤𝑘 and ℎ≤𝑘. By the second axiom, an element 𝑓∈𝐷(𝑋) cannot be decomposed into any other element of 𝐷(𝑋) if and only if, it is an element of the free commutative semigroup Irr(𝑋)^⊕ generated by the irreducible elements, that is, a finite linear combination of irreducible elements (with nonnegative integer coefficients). Hence, when 𝐷(𝑋) is refinable, for each 𝑥∈𝑋∩𝐷(𝑋), there is at most one way to decompose 𝑥 into irreducibles (that is to fully decompose 𝑥). In terms of rewriting systems, it is known as the property of confluence. Finally, if 𝐷(𝑋) is both refinable and finite, then any 𝑥∈𝑋 as a unique decomposition into irreducibles. This is precisely the property of convergence of a (Noetherian and confluent) rewriting system. Let 𝐺 be any Abelian group, and 𝐷(𝑋) be any decomposition of 𝑋. A mapping 𝑓∶𝑋→𝐺 is said to be an invariant when for every 𝑥≤∑^𝑘_𝑖=1𝑛_𝑖𝑥_𝑖 in 𝐷(𝑋), we have 𝑓(𝑥)=∑^𝑘_𝑖=1𝑛_𝑖𝑓(𝑥_𝑖). We are now in position to state Brylawski’s main result (to compare to Theorem 16).

Theorem 18 (see [3]). Let 𝐷(𝑋) be a finite and refinable decomposition of 𝑋. There exist an Abelian group 𝐴 and an invariant mapping 𝑡∶𝑋→𝐴 such that for every Abelian group and every invariant mapping 𝑓∶𝑋→𝐺, there exists a unique group homomorphism ℎ∶𝐴→𝐺 such that the following diagram commutes: (A.1) Moreover, 𝐴 is freely generated by Irr(𝑋).