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Grothendieck Ring of the Pairing Function Without Cycles

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Grothendieck Ring of the Pairing Function Without Cycles Grothendieck ring of the pairing function without cycles Esther Elbaz February 12, 2020 Abstract A bijection (l; r) between M 2 and M is said to be a pairing function with no cy- cles, if any composition of its coordinate functions has no fixed point. We compute here the Grothendieck ring of the pairing function without cycles to be isomorphic to 2 2 Z ' Z[X]=(X − X ). In a previous article, we illustrated ideas to construct, given a quotient R of a polynomial ring over Z, a structure whose Grothendieck ring is Z. Following these very ideas, the 2 structure we should consider to obtain a structure with Grothendieck ring Z[X]=(X − X ) is precisely the pairing function without cycles. The method we use here to compute its Grothendieck ring has already allowed us to treat the case of a bijection, without cycles, between a set and itself deprived of N point, for any N 2 N (the Grothendieck ring is Z[X]=(N)), and will allow us to treat in forthcoming papers the bijection without cycles between a set and the union of two disjoint copies of itself (the Grothendieck ring is Z), and for every integer N, an enrichment of this structure whose Grothendieck ring is Z=NZ. Contents 1 Introduction1 1.1 Method........................................3 1.2 Pairing function without cycles...........................5 2 Representation of formulas and definable sets by binary trees6 3 Closed tree and simple formulas9 4 Form of the definable injections 11 5 Simple sets 13 6 Decomposition of definable sets 14 7 Definable injection on simple set 16 8 Representation of a decomposition: tree of a decomposition 17 9 Computation of the Grothendieck ring 20 10 Example of pairing function without cycles on N 22 1 1 Introduction The notion of Grothendieck ring of a theory has been introduced in the early 2000. F. Loeser an J. Denef [LD] on the one hand introduced the notion of the Grothendieck ring of a theory through motivic integration. T. Scanlon and J. Krajicek [SK] on the other hand considered the Grothendieck ring of a structure and built up a dictionary between the geometric properties of the Grothendieck ring of a structure, and the combinatorial properties of this structure. Definition 1.1. Let M be a non empty structure and let Def(M) be the set of definable (with parameters) subsets of some Cartesian product of M. We define an equivalence relation on Def(M) by: A; B 2 Def(M) are equivalent if and only if, there exists a definable bijection between them. We consider the quotient set, denoted Def(g M), and denote by [A] the class of any subset A of Def(M). On this quotient set, we define two laws: • An additive law defined by: [A] + [B] = [A \ B] + [A t B]. • A multiplicative defined by: [A] ∗ [B] = [A × B]. The additive law corresponds on the level of sets to the disjoint union and admits the class of the empty set as neutral element. The multiplicative law corresponds on the level of sets to the Cartesian product and admits the class of a singleton as neutral element. The additive law is not cancellative: a + b = a + c doesn’t imply that b = c. In order to make it cancellative, we quotient Def(g M), by the equivalent relation defined by: a; b 2 Def(g M) are equivalent if and only if there exists c 2 Def(g M) such that a + c = b + c. We then obtain a cancellative monoide for the additive law. There exists a unique (up to isomorphism) ring that embeds this last quotient. This ring is called the Grothendieck ring of M and is denoted K0(M). The Grothendieck ring satisfies the following universal property: let f be a f+; ×}-homomorphism from Def(g M) to a ring. Then f factors through K0(M). As an example, one can easily checked that the Grothendieck ring of any finite structure is Z: the class of a definable set in the Grothendieck ring will correspond to the class of its cardinality. The real closed field, R also admits Z as its quotient. It has been demonstrated by J. Krajicek and T. Scanlon [SK] and relies on the fact that two sets that have the same dimension and the same geometric Euler characteristic constructed on the definable subsets of R through triangulation (see for example [VD]) are in definable bijection. The geometric Euler characteristic is s a result the quotient map that associates to any definable subset its class in the Grothendieck ring.In particular, it does not account for the dimension of the definable <0 F F >0 subsets. For example R is equal to the disjoint union R f0g R and, in the language of the fields, both R>0; R<0 are in definable bijection with R. Noting [R] the class of R in its Grothendieck ring, we obtain that 2[R] + 1 = [R] which implies that [R] = −1 : the class of R is equal to the opposite of the class of a singleton. Thanks to the second quotient, we can easily prove a property, called the onto-pigeonhole principle and abbreviated onto-PHP, which states that the Grothendieck ring of a structure is non-zero if and only if none of its definable sets is in definable bijection with itself deprived of a point. 2 This property demonstrated by J. Krajicek and T. Scanlon in [SK] is an example of the par- allel between the combinatorial properties of a structure and the algebraic properties of its Grothendieck ring.It has been used to show the triviality of the Grothendieck ring of several rings and fields, like for example, of Laurent series fields [?], or more generally of certain valued fields in Z [?, ?]. In this article, we compute the Grothendieck ring of the pairing function without cycles to be isomorphic to Z2 ≡ Z[X]=(X − X2). In [?], we illustrated a method to construct given R, a fixed polynomial ring over Z, a structure that admits R as its Grothendieck ring. To do so we consider a structure with exactly the functions that are require to obtain this Grothendieck ring . Indeed, noting X the class of a structure M in its Grothendieck ring, the relation X = X2 precisely means that M is in definable bijection with M 2. The property of being without cycles is, in ad-equation with the general ideas explained below, required to get quantifier elimination. In the next subsection, we give an idea of what this method consists of. We then apply it to the pairing function without cycles and prove our theorem. Theorem 1.2. Let L := fl; rg be a language with two unary function. Let M be a L-structure such that the 2-ary function (l; r) is a bijection between M and M 2. Assume furthermore, that this pairing function has no cycles: any function f obtained as a composition of the function fl; rg has no fixed point. 2 Then K0(M) is isomorphic to Z . 1.1 Method We use the convention N = Z≥0. Let R be any quotient of Z[X]. We have explained in [?] how one can construct a somehow minimalistic theory such that any model of this theory will admit R as its Grothendieck ring.We recall in this section, the basic ideas. In order to construct a structure admitting a prechosen ring, we proceed in two steps. First, we design the structure. Second, we compute its Grothendieck ring and thus show that it is indeed the one we are expecting. Remark 1.3. We should highlight that, in the general case, the Grothendieck ring of a structure is not deter- mined by its theory. Two elementary equivalent structures do not necessarily have elementary equivalent Grothendieck ring (this is due to the fact that we consider set definable with param- eters). It has been proven [SK] that if M ≡ N, then K0(M) ≡91 K0(N) and, later, that if N is an elementary extension of M, we have an injection of its Grothendieck ring into K0(M). In the case of the theory of modules, S. Perera [?] proved that all of their elementary equivalent models have elementary isomorphic Grothendieck rings. But this does not hold in general. The theories of our construction have the particularity that any of their model has the same Grothendieck ring. The ring R being a quotient of Z[X], there exists an ideal I of Z[X] such that R = Z[X]=I. It is well known that any ideal of Z[X] admits a generating subset consisting of a polynomial P (X) (possibly a constant polynomial) and an integer (possibly zero). We can show that if we 3 can construct a structure whose Grothendieck ring is R for any R generated by a polynomial over N and possibly a constant, then we can treat the general case. This results from the following lemma: Lemma 1.4. Let Q(X) 2 Z[X] whose leading coefficient is positive. Then there exists P (X) 2 N[X] and k 2 N such that P (X − k) = Q(X). Let M be a structure whose Grothendieck ring is Z[X]=(P (X)) (respectively Z[X]=(N; P (X)) where N 2 N). Then there exists a structure N whose Grothendieck ring is Z[X]=(Q(X)) (respectively Z[X]=(N; Q(X)) where N 2 N). Proof. We won’t prove this lemma in detail here but just give the general idea. It is easy to prove the first part.
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