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

Laurent Poinsot

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.

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 and a polynomial in two commuting variables , 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 be a graph, and let be an edge of , 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 of is the graph obtained by identifying the vertices source and target of , and removing the edge . We write for the graph where the edge is merely removed; this operation is the edge deletion. Let us consider the graph (well-defined as isomorphic classes) which can be interpreted as a decomposition of . Then, the Tutte polynomial is invariant with respect to this decomposition in the sense that . 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 be a set, and let be an order relation on (a part of) the free commutative semigroup (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 be a set of formal (finite) sums , where , not all of them being zero (an element of the free commutative semigroup on ) partially ordered by . If such that , then we say that   decomposes into or that is a decomposition of . Therefore is seen as a set of commutative decompositions. Elements of that belong to are assumed to be minimal with respect to . Elements of 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 cannot be decomposed further into any other element of if and only if, is a finite linear combination, with nonnegative integers as coefficients, of incomparable elements, that is, if is the set of all irreducibles, then is not decomposed into another element if and only if, is a formal (finite) sum of elements of with nonnegative integers as coefficients. This property is similar to the notion of termination in rewriting systems. Two other properties (refinability and finiteness) on ensure that every element of has one, and only one, “terminal” decomposition into irreducible elements. They are equivalent to convergence of a rewriting system. For instance, the order on the free commutative semigroup generated by all (isomorphism classes of) finite graphs satisfies these axioms and properties.

Now, to a decomposition with the above properties may be attached a group in a universal way. A function from to an Abelian group is said to be invariant if for every such that (, and ), then . Recall here that Tutte polynomial is invariant because . Brylawski proved the following theorem, which was his main result. There exist an Abelian group, called Tutte-Grothendieck group, and an invariant mapping , called universal Tutte-Grothendieck invariant, such that for every Abelian group and every invariant mapping , there exists a unique group homomorphism with . In addition, is isomorphic to the free Abelian group with the irreducible elements as generators. In the classical context of graph theory, as expected, is the additive structure of and 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 , , and 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 , , and 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 , , and , respectively, the free semigroup, monoid, group generated by (see [5]), and we identify as a subset of each of these algebraic structures. Note also that we denote by the free commutative semigroup on .

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

The forgetful functor from 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 be a monoid. Then there exists a unique group , called the group completion or universal enveloping group or Grothendieck group of (see [6] and references therein, and also [7]), and a unique homomorphism of monoids such that for every group and every homomorphism of monoids , there is a unique homomorphism of groups such that the following diagram commutes (in the category of monoids):

(1)

It is not difficult to check that is given either as , where is the subset (where “” is the monoid multiplication of , and where denotes the free group of , see Section  2.1 and if is a group and is any subset of , then is the normal subgroup of generated by ), see [7], or as the quotient monoid where (here the star “” stands for the free monoid functor, see also Section  2.1, and is the empty word) where is the set of (formal) symbols equipotent to .

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 so that all statements make sense in any of these categories.

Let be a set and let be a symmetric (i.e., for every , implies ) and irreflexive relation on (i.e., for each , ). Let be an object in , and be a set-theoretical mapping. This function is said to respect the commutations whenever then , for every . A pair is called a commutation alphabet.

It can be shown that there exists a unique object of and a unique mapping that respects the commutations such that for every object of and every mapping that respects the commutations, there is a unique arrow (in ) such that the following diagram commutes in the category of sets:

(2)

The object is usually called the free partially commutative semigroup (resp., monoid, group) on (or on to be more precise) depending on , and may be constructed as follows: and where is the congruence on or generated by whenever for all (the least congruence on or containing the relation whenever for all , see [11]), and .

We may note that is nothing else than the usual free (noncommutative) object in the category , while , where is the equality relation on , is the free commutative object in (in particular, 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 is isomorphic to (actually , where is the empty word) in such a way that embeds in as a subsemigroup.

Lemma 1. The monoid is isomorphic to the free adjunction of an identity to the semigroup .

Proof. To prove this lemma it is sufficient to check that is a solution of the universal problem of adjunction of a unit to . According to the universal problem of the free partially commutative semigroup , there is a unique homomorphism of semigroups such that the following diagram is commutative:

(3)

Now, let be a monoid and be a semigroup homomorphism. Therefore there exists that respects the commutations and such that . According to the universal problem attached to , there is a unique homomorphism of monoids such that . Therefore, , but then . The relations between all the arrows are summarized in the following commutative diagram:

(4)

There is also an important relation between and given in the following lemma.

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

Proof. The set-theoretical mapping respects the commutations, therefore according to the universal problem of the free partially commutative monoid over there is a unique homomorphism of monoids that makes commutes the following diagram:

(5)

Now, let be any group, and be an homomorphism of monoids. Then, according to the universal problem of the free partially commutative monoid, there is a unique set-theoretical mapping that respects the commutations and . Now according to universal problem of , is uniquely extended as a group homomorphism such that . Therefore, so that (by uniqueness of a solution of a universal problem). Therefore is a solution of the universal problem of the group completion of . 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 of the proof of Lemma 2 is one-to-one so that may be identified with a submonoid of its Grothendieck completion .

Definition 3. Let be any set. For every and every , let us define as the number of occurrences of the letter in the word . More precisely, if is the empty word, then , if , if for all , and if the length of is >1, then for some letter , and , then . Let be a congruence on or . It is said to be multihomogeneous if for every in or , such that , then for every , . Therefore we may define for the class of modulo it does not depend on the representative of the class modulo .

According to [10], any congruence of the form is a multihomogenous congruence, so that we may define for all and all (where or ). The notion of multihomogeneity is used to check that we may identify the alphabet as a generating set of using the map , which is shown to be one-to-one, in such a way that we consider that . Indeed, for semigroup or monoid case, let such that their classes modulo be equal. But is a multihomogenous congruence (see [10]). Therefore . Concerning the group case, let us assume that are equivalent modulo the normal subgroup so that there is some with . Because the group is free, it means that (no nontrivial relations between the generators). In the sequel, we will treat as a subset of .

More generally, let be a commutation alphabet and let . We define . It is possible to embed into as illustrated in the following lemma.

Lemma 4. Under the previous assumptions, there is an arrow in the category which is one-to-one.

Proof. Let be the canonical inclusion. Define as the unique arrow (in ) such that the following diagram commutes:

(7)

Therefore, .

Let . Let us define such that for every , and for . We note that . Then we may consider as the unique arrow (in ) that makes commutes the following diagram:

(8)

Therefore . Now, , so that (by uniqueness) , and then is one-to-one (and is onto).

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

Lemma 5. Let be a commutation alphabet, and let be any subset. Let . The following statements are equivalent.

(1). (2)For all , implies that .

Proof. Let . If , then for all such that , (since ). Because is a multihomogeneous congruence, for all and . Then the point . is obtained. Now, let such that for all , implies that . Then, for all (), for all which means that , and therefore so that 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 be a set, and be any binary relation, called a (one-step) reduction relation, and is called an abstract rewriting system. We denote by “” the membership “”, and “” stands for “”. Let be the reflexive transitive closure of a binary relation . We use or to mean that or . An element is said to be reducible if there exists such that . is irreducible if it is not reducible, or, in other terms, if is -minimal: there is no such that . A normal form of is an irreducible element such that . If it exists and is unique (see below), the normal form of is denoted by . The set of all normal forms, or equivalently, of all irreducible elements is denoted by or when this causes no ambiguity. Note that two distinct normal forms are -incomparable, that is and . A reduction relation is said to be terminating or Noetherian if there is no infinite -descending chain of elements of such that for every . In particular, if is terminating, then it is irreflexive (otherwise for some such that 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 () is terminating or Noetherian whenever is so. Two elements are said to be joinable if there is some such that , and (and also ()) is said to be confluent if for every such that , then are joinable. A reduction relation , and an abstract rewriting system , 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 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 containing ), then if and only if, , therefore satisfies and so is onto and moreover, the function which maps the class of modulo to is well defined, onto and one-to-one.

3.2. Semigroup Rewriting Systems

Now, let us assume that is actually a semigroup . Let be any binary relation. We define the following relation by if and only if, there are and such that and . A relation is called the (one-step) reduction rule associated with . A relation is said to be two-sided compatible if () implies . Now, the intersection of the family of all two-sided compatible relations containing a given (this family is nonvoid since it contains the universal relation ) also is a two-sided compatible relation, and so we obtain the least two-sided compatible relation that contains . It is called the two-sided compatible relation generated by , and it can be shown that this is precisely . Now, given , is called a (semigroup) rewriting system; definitions and properties of an abstract rewriting system may be applied to such a rewriting system. When is the free monoid , 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 of is actually a semigroup congruence, because is two-sided compatible. The quotient semigroup is called the Thue semigroup associated with the semigroup rewriting system .

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 be a commutation alphabet, and . Then 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 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 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 . Then, . As a result, is a subsemigroup of .

Proof. Let us assume that . Therefore there are , , such that , and (so that ). Because is multihomogeneous, either or for some . But in this case, either or is reducible, which is a contradiction. As a result, is closed under the operation of so that is a subsemigroup of .

Corollary 8. The map is a surjective homomorphism of semigroups.

Proof. Let . According to Lemma 7, . Therefore, . Since is a congruence of , in such a way that and is an homomorphism of semigroups. It is obviously onto.

Corollary 9. The semigroups and are isomorphic.

Proof. As introduced in Section  3.1, let be the function that maps the class of modulo to the normal form . It is a one-to-one and onto set-theoretical mapping. But according to Corollary 8, is a semigroup homomorphism, in such a way that also is.

The fact that the rewriting system is alphabetic (Definition 6) actually implies that the (isomorphic) semigroups and 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 (recall from Section  2.3 that is considered as a subset of ). It is clear that . Indeed, for every , , if and only if, there are , , such that and . Since is a multihomogenous congruence (see Section  2.3), is the empty word, and , . Therefore if and only if, .

This characterization of is used in the following lemma.

Lemma 10. If , then .

Proof. Let us assume that and . Let . Since , there is some such that . Because , and generates , it can be written as for some , and . Because , there is such that . Then, . Replacing by , we may construct an infinite descending chain , which is impossible since is assumed to be convergent, and therefore terminating. So .

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

The following lemma reveals the structure of free partially commutative semigroup of , and therefore also of according to Corollary 9.

Proposition 12. The semigroup is equal to the free partially commutative semigroup where (see Lemma 4).

Proof. Let . Let us assume that . According to Lemma 5, there exists such that for all , ( is seen as a congruence class), . Therefore for some and (where is the canonical epimorphism and where we recall that is seen as a subset of , , and ). But , then there exists such that , and therefore which contradicts the fact that . Let such that . Let us assume that . Therefore for some , such that there is with . Therefore . It is then clear that for every such that , . But according to Lemma 5, this is impossible because and . We have proved that and are equal as sets. But since they are both subsemigroups of , then they are equal as semigroups.

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

Definition 13. Let be a commutation alphabet, and let be an alphabetic rewriting system. Let be any semigroup, and let that respects the commutations. Let be the unique homomorphism of semigroups such that the following diagram commutes (see Section 2.3):

(9)

Then is said to be an -invariant if for every and such that , then .

Informally speaking, according to Definition 13, a function that respects the commutations is an -invariant if its canonical semigroup extension is constant for all reductions .

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

Lemma 14. Let be a semigroup, and let be a function that respects the commutations. Let be its canonical semigroup extension from to . If is a -invariant, then for every such that , one has .

Proof. Since we will deal with the empty word, one needs to recall the following. According to Lemma 1, , where is the empty word. Let us define the canonical extension of as a monoid homomorphism. That is, whenever , , and . Let such that . Then there exist , , such that , and . Because is -invariant, , and then we have .

Corollary 15. Let be a semigroup, and let be a function that respects the commutations. If is a -invariant, then its canonical semigroup extension factors through the quotient .

Proof. Let such that . Then there are , , , such that for every , or . Therefore for every , either , or , or . Because is a -invariant, according to Lemma 14, for every , . Therefore . Then, there exists a unique semigroup homomorphism such that for every (where denotes the class of modulo ).

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

Theorem 16. Let be a commutation alphabet, and let be an alphabetic rewriting system. There exist a group and a mapping that respects the commutations which is an -invariant such that for every group , and every (commutations respecting) -invariant mapping , there is a unique group homomorphism such that the following diagram commutes:

(10)

Moreover, if is convergent, then the group is isomorphic to the free partially commutative group and is the normal form restricted to the alphabet (recall that one has under natural identifications; see Section  2.3).

Proof. Let be a group and let be a commutations respecting -invariant mapping. According to the universal problem of free partially commutative semigroups, because 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 as a monoid homomorphism (because is also a monoid). Let us denote by the monoid . We obtain the following diagram:

(13)

Finally, using the Grothendieck group of , we complete the previous commutative diagram as follows (where ):

(14)

Now, as illustrated in the previous diagram, let , , and . First of all, obviously respects the commutations. Let us consider the canonical extension of . 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 . Now, let , such that . Then, . Therefore, , so that is -invariant. Then the first part of the theorem is proved.

Now, let us assume that is convergent. Then, by Proposition 12, is isomorphic to the free partially commutative semigroup . Therefore, is isomorphic to the free partially commutative monoid (by Lemma 1). Finally, the Grothendieck group is isomorphic to the Grothendieck group (because is functorial) so that it is isomorphic to the free partially commutative group (by Lemma 2). The fact that in this case, is the normal form restricted to the alphabet (where is naturally identified with a subsemigroup of ) is quite obvious to check.

Definition 17. The group is called the Tutte-Grothendieck group and the universal Tutte-Grothendieck -invariant of the alphabetic rewriting system .

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 be a multigraph, and be a link (edge which is not a loop nor a bridge) in . Let be the graph obtained from by erasing , and let be the graph obtained by contraction of in ( is removed, and its origin and source are identified). Then is decomposed into (+ 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 , let be the free commutative semigroup generated by (i.e., , where and is the equality relation on ), written additively. Recall also that the free Abelian group generated by , namely , is isomorphic to the group (under point-wise addition) of all mappings from to with a finite support (the support of a function is the set of all such that ), see for instance [5]. Let be a two element set. Let , and . It is clear that . Moreover the alphabetic rewriting system is convergent (it is not difficult to check this property using for instance techniques from [17]). Let . Then . Therefore we recover the well-known fact (see [18]) that the integral Weyl algebra with two generators (where denotes the ring of the free monoid , and where is the two-sided ideal of generated by ) is free as an Abelian group with generators . The universal Tutte-Grothendieck -invariant of is the normal form of the words in . For instance, .

Let be a variable (distinct from ) and let . Consider the relation . Finally let . Therefore commutes with all elements of . Let . Then we can check that is a convergent alphabetic rewriting system whose Tutte-Grothendieck group is where (note that for every nonnegative integers , , , , and , since commutes with all other elements). This gives us immediately a free -basis for the central extension (where is the two-sided ideal of the ring of the monoid generated by and for every ) of the integral Weyl algebra .

4.3.3. Integral Temperley-Lieb Algebra

Let such that . Let us consider . Let ,. Let . Let ,. This alphabetic rewriting system is easily shown to be convergent. The corresponding Tutte-Grothendieck ring is . We recover the fact that the -th Temperley-Lieb -algebra (with parameter ) , where the two-sided ideal generated by the following relations for , for , for is free with basis .

4.3.4. The Poincaré-Birkhoff-Witt Theorem

Let be a Lie algebra over which is free as an -module (see [19]). Let be a basis of seen as a (free) -module. Let us assume that is linearly ordered by . Let be the free monoid generated by . Let . It is obvious that is a convergent alphabetic rewriting system. Moreover, for all and its Tutte-Grotendieck group is , while its universal Tutte-Grothendieck invariant is the reordering of an element of in an increasing order (relative to ). We recognize the famous Poincaré-Birkhoff-Witt theorem ([20–22]) since the universal enveloping -algebra of over is free with basis .

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 be a set together with a multivalued binary operation (meaning that implies that ) subjected to properties given below. For every , is a finite set. The operation is extended to the power set of by for some . If and , then we let be equal to , and is defined by induction: , and for every positive integer . We say that is a prefab if the composition on is associative, commutative (therefore becomes a semigroup), and has an identity (the identity plays also a role in counting arguments in [23]) such that for every (then is a monoid). An element is called a prime if implies or . We say that is a unique factorization prefab if every factors uniquely into primes in the sense that for a unique set of primes and unique positive integers . We say that is a very unique factorization prefab if where ,  , all the ’s and all the ’s are positive integers, all the ’s are mutually distinct primes, and all the ’s are mutually distinct primes (but some ’s may be equal to some ’s), then there exist unique elements and such that . In the original definition of a prefab, there is also a mapping which serves as a weigth function for a combinatorial use but which is not needed here.

Let be a unique and very unique factorization prefab. Let be the set of primes of this prefab. Let . Let . According to the properties of unique factorization, very unique factorization, associativity, and commutativity of , it is clear that is a convergent alphabetic rewriting system. We have , and the Tutte-Grothendieck group is, as expected, . It is also immediate that where is the unique prime factorization of .

As examples of (unique and very unique factorization) prefabs, one can cite the following two from [23]. Let be any set, and let be two words. A shuffle of these two words is a word , for (where is the sum of lengths of and ) such that there exists a partition of with , , is the cardinal of , , and , , is the cardinal of , (such constructions appear in the shuffle product of two words; see [24]). Let be the set of all shuffles of and . As an example, and . Then . It is clear that the identity is the empty word (therefore we allow to choose word in ) while the prime elements are the letters in . 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: where such that . To summarize, the set is , the Tutte-Grothendieck group is , and the universal invariant is given by (which is sometimes called the commutative image; see [25]).

Let be an indeterminate for each such that whenever . Let . Let except a finite number, . Finally let us define . For instance, . The identity is , while the primes are exactly the for , where is the set of all prime integers. Attached with these datas, the rewriting system on is given by . The Tutte-Grothendieck group is , and the universal invariant is given by , where is the decomposition of 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 be a set, and let . Let be a partially ordered set with such that

(1)for every , if , then or (where ), (2)for every , if and only if, there exist in such that , , and .

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

Theorem 18 (see [3]). Let be a finite and refinable decomposition of . There exist an Abelian group and an invariant mapping such that for every Abelian group and every invariant mapping , there exists a unique group homomorphism such that the following diagram commutes:

(A.1)

Moreover, is freely generated by .

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