Jean-Pierre Magnot

ON MATHEMATICAL STRUCTURES ON PAIRWISE COMPARISONS MATRICES WITH COEFFICIENTS IN A GROUP ARISING FROM QUANTUM GRAVITY

Keywords: approximate reasoning, inconsistency, pairwise comparisons, group, holonomy, matrix, simplex, graph. 2010 Mathematics Subject Classification: 03F25

We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example where the use of a non abelian group is necessary. Algebraic, topological, geometric and probabilistic aspects are considered.

Introduction

Pairwise comparisons are among the classical ways of decison making and information checking. The main idea is simple: assign a score to the comparison of a pair (s, s ) of two states which have to be compared, and we say that the "scores" are consistent if, for three states (s, s , s ), the comparison of s and s can be deduced from the comparisons of s and s , and of s and s . If not, the comparisons are called inconsistent. These aspects are precised in section 1, and gives so many applications that it is impossible to cite them all. We mention two of them [START_REF] Cavallo | Pairwise Comparison Matrices over abelian Linearly Ordered Groups: A Consistency Measure and Weights for the Alternatives[END_REF][START_REF] Crawford | The analysis of subjective judgment matrices; A project AIR FORCE report prepared for united states air force report[END_REF][START_REF] Lahby | Network Selection Decision based on handover history in Heterogeneous Wireless Networks[END_REF] which are applications of deep interest. This is mostly why some authors have developed ways to quantify inconsistency [START_REF] Koczkodaj | A new definition of consistency of pairwise comparisons[END_REF][START_REF] Saaty | A scaling methods for priorities in hierarchical structures[END_REF], see e.g. [START_REF] Bozóki | On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices[END_REF], and there exists actually tentatives of axiomatizations of the so-called "inconsistency indicators" [START_REF] Brunelli | Axiomatic properties of inconsistency indices for pairwise comparisons[END_REF][START_REF] Brunelli | Studying a set of properties of inconsistency indices for pairwise comparisons[END_REF][START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons Matrices Revisited[END_REF]. One can also try to deal with partial orders, after [START_REF] Janicki | Pairwise Comparisons Based Non-Numerical Ranking[END_REF][START_REF] Janicki | A weak order solution to a group ranking and consistencydriven pairwise comparisons[END_REF][START_REF] Janicki | On a paiwise-based consistent non-numerical ranking[END_REF][START_REF] Zhai | Non numerical ranking based on pairwise comparisons PhD thesis[END_REF], motivated by the obvious lack of informations when one tries to express a complex situation only by a , or by the necessary hierarchization of constraints in a difficult problem.

This leads us to the main motivation of this work. Dealing with partial orders can turn out to be very complex (see e.g. [START_REF] Zhai | Non numerical ranking based on pairwise comparisons PhD thesis[END_REF]), where as (non abelian) groups furnish a minimal setting where composition and inversion are well-defined, with all the necessary properties for comparisons of more complex datas. Such an approach is already used in [START_REF] Farinelli | Geometric arbitrage theory and Market dynamics[END_REF][START_REF] Farinelli | Gauge invariance[END_REF][START_REF] Illinski | Gauge geometry of financial markets[END_REF][START_REF] Young | Foreign exchange markets as a lattice gauge theory[END_REF], linked with so-called gauge theories in quantum physics [START_REF] Magnot | A mathematical bridge between discretized gauge theories in quantum physics and approximate reasoning in pairwise comparisons[END_REF][START_REF] Young | Foreign exchange markets as a lattice gauge theory[END_REF]. Other applied approaches can be found in [START_REF] Mack | Gauge theory of things alive[END_REF][START_REF] Ramik | Pairwise comparison matrix with fuzzy elements on alo-group[END_REF][START_REF] Sengupta | Towards a Neuronal Gauge Theory[END_REF][START_REF] Tozzi | A timeless biology[END_REF] which shows the great vitality of this topic for applications in various fields. The aim of this paper is to draw-back the structures highlighted by physics and to adapt them in the context of pairwise comparisons matrices. As a by-product, the choice of the largest possible setting is natural, since we intend here to describe new perspectives of methods, even if most of pairwise comparisons matrices have coefficients in R * + . This is also the occasion to analyze whether classical objects in pairwise comparisons 1 matrices arise from higher mathematical or physical considerations, or if they are only valid in the R * + -setting. Let us now describe the contents of this paper. Section 1 reviews selected topics on R * + -pairwise comparisons matrices which will give rise to natural generalizations linked with a quantum physics picture.

Section 2 describes pairwise comparisons matrices with coefficients in a group G, indexed by any (finite or infinite) set of states I. We define also what is an inconsistency map. The terminology of inconsistency indicator is reserved to inconsistency maps with additionnal properties along the lines of [START_REF] Koczkodaj | On normalization of inconsistency indicators in pairwise comparisons[END_REF], and the settings developed will be justified by other parts of this work.

Section 3 deals with algebraic properties of consistencywhich extends as naturally as possible the classical setting recovered setting G = R * + . We highlight adjoint action of the group G I , called gauge group by analogy with differential geometric settings [START_REF] Koczkodaj | A Geometric Framework for the Inconsistency in Pairwise Comparisons[END_REF], on the set of pairwise comparisons matrices. The key property states that consistency is characterized by an orbit of this group. The side-properties are then described, which motivates the vocabulary for properties of inconsistency maps, and leads to the terminology of inconsistency indicator. As a concluding property, we show that Koczkodaj's inconistency map is an adjoint-invariant inconsistency indicator. We are aware that a similar study, based on a deep understanding of the properties of Saaty's inconsistency map [START_REF] Saaty | A scaling methods for priorities in hierarchical structures[END_REF], is actually investigated by others. Section 4 deals with generalization on graphs. This happens when two states cannot be compared by direct comparisons, but only by comparisons with intermediate states. This leads to "holes" in the pairwise comparisons matrices, assigned to the coefficient "0", and the notion of holonomy enables us to extend Koczkodaj's inconsistency maps to a R[[X]]-valued inconsistency map, which is adjoint-invariant.

In section 5, we first answer to [START_REF] Koczkodaj | Inconsistency indicator maps on groups for pairwise comparisons[END_REF] where an approach by so-called G-distances is prposed. We feel the need to show that this approach does not seem natural for us an from the viewpoint of this work. More natural is the topological notion of filter. We show how inconsistency maps naturally give rise to a filter of so-called vincinities of the set CP C I (G). We cannot call it "filter of neighborhoods" because the notion of neighborhood is a very precise notion in topology, linked with the notion of continuity of maps. Here, the inconsistency indicators are not a priori continuous, this is why we feel the need to highlight the distinction by the chosen vocabulary.

In section 6, motivated by Yamabe theorem that we recall briefly, we analyze sme geometric aspects of pairwise comparisons. The correspondence with gauge theories then arises clearly. The objectives of this section are as follows. We consider the conditions of consistency and inconsistency in the geometric setting of a finite or infinite dimensional simplex (section 6.1), which can be understood as higher dimensional triangles (which are 2-simplexes) and tetrahedra (which are 3-simplexes), see e.g. [START_REF] Whitney | Geometric Integration Theory[END_REF][START_REF] Friedman | Survey article: an elementary illustrated introduction to simplicial sets[END_REF]; each s i corresponds to a 0-vertex; each 1-vertex gives an edge, and a triad (where inconsistency can be measured) is a 2-face ( or 2-simplex). According to this setting, a geometric picture which is very similar to inconsistency is the holonomy of a connection (see, for example, [START_REF] Kolar | Natural operations in differential geometry Springer[END_REF] for holonomy in finite dimensions, and [START_REF] Magnot | q-deformed Lax equations and their differential geometric background Lambert[END_REF] for the infinite dimensional case). We show that a PC matrix A can always be expressed as a holonomy matrix if the group G is exponential, and when G = (R * + ) J , consistent PC matrices are holonomies of a flat connection which can be constructed.

We finish with probabilistic approaches and examples. Examples under consideration may be qualified as toy examples, given to give accessible situations which mathematical intuition can be compared with. Necessity of strongest examples seem unnecessary in view of the literature given in bibliography where simplicial or lattice gauge theories (i.e. pairwise comparisons) arise. Concerning probabilistic aspects, we restrict ourselves to interpret, in terms of pairwise comparisons matrices, two probabilistic approaches of gauge theories. The first approach presented relies cylindrical approximations of the Weiner measure, while the second approach intends to explain in a way as simple as possible a way to understand inconsistency indicators in a way parallel to Lagrangian theories and action functionals. These very technical aspects of mathematics used in physics are here simplified and adapted in the spirit of the whole paper.

Pairwise comparisons matrices with coefficients in R +

It is easy to explain the inconsistency in pairwise comparisons when we consider cycles of three comparisons, called triad and represented here as (x, y, z), which do not have the "morphism of groupoid" property such as

x.z = y Evidently, the inconsistency in a triad (x, y, z) is somehow (not linearly) proportional to y -xz. In the linear space, the inconsistency is measured by the "approximate flatness" of the triangle. The triad is consistent if the triangle is flat. For example, (1, 2, 1) and [START_REF] Farinelli | Geometric arbitrage theory and Market dynamics[END_REF]101,[START_REF] Farinelli | Geometric arbitrage theory and Market dynamics[END_REF] have the difference y -xz = 1 but the inconsistency in the first triad is unacceptable.It is acceptable in the second triad. In order to measure inconsistency, one usually considers coefficients a i,j with values in an abelian group G, with al least 3 indexes i, j, k. The use of "inconsistency" has a meaning of a measure of inconsistency in this study; not the concept itself. The approach to inconsistency (originated in [START_REF] Koczkodaj | A new definition of consistency of pairwise comparisons[END_REF] and generalized in [START_REF] Duszak | Generalization of a New Definition of Consistency for Pairwise Comparisons[END_REF]) can be reduced to a simple observation:

• search all triads (which generate all 3 by 3 PC submatrices) and locate the worse triad with a so-called inconsistency indicator (ii), • ii of the worse triad becomes ii of the entire PC matrix.

Expressing it a bit more formally in terms of triads (the upper triangle of a PC submatrix 3 × 3), we have:

(1.1) Kii(x, y, z) = 1 -min y xz , xz y .
According to [START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons[END_REF], it is equivalent to:

ii(x, y, z) = 1 -e -|ln( y xz )|

The expression | ln( y xz )| is the distance of the triad T from 0. When this distance increases, the ii(x, y, z) also increases. It is important to notice here that this definition allows us to localize the inconsistency in the matrix PC and it is of a considerable importance for most applications.

Another possible definition of the inconsistency indicator can also be defined (following [START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons[END_REF]) as:

(1.2) Kii n (A) = 1 -min 1≤i<j≤n min a ij a i,i+1 a i+1,i+2 . . . a j-1,j
, a i,i+1 a i+1,i+2 . . . a j-1,j a ij since the matrix A is consistent if and only if for any 1 ≤ i < j ≤ n the following equation holds:

a ij = a i,i+1 a i+1,i+2 . . . a j-1,j . It is equivalent to: (1.3) Kii n (A) = 1 -max 1≤i<j≤n 1 -e -ln a ij a i,i+1 a i+1,i+2 ...a j-1,j
The first Koczkodaj's indicator Kii 3 allows us not only find the localization of the worst inconsistency but to reduce the inconsistency by a step-by-step process which is crucial for practical applications. The second Koczkodaj's indicator Kii n is useful when the global inconsistency indicator is needed for acceptance or rejection of the PC matrix. An abstract unification will be proposed at the end of section 4.

Changing the comparisons structure to arbitrary groups

In the previous section, the comparisons coefficients are a i,j are scaling coefficients. This means that,if the PC matrix A is coherent, given a state s k , we can recover all the other states s j by something assimilated to scalar mutiplication:

s j = a j,k s k .
In other words, even if the states s j are driven by more complex rules, we reduce them to a "score" or an "evaluation" in R + . This is useless to say that such an approach is highly reductive: even in video games, the virtual fighters have more than one characteristic: health, speed, strength, mental... and the global design of these characteristics intends to reflect some "complexity" in the game (please note the " "). So that, the states s j have to belong to a more complex state space S, and in order to have pairwise comparisons, a straightforard study shows that we define [START_REF] Lawvere | Conceptual Mathematics: A First Introduction to Categories[END_REF][START_REF] Lane | Categories for the Working Mathematician Graduate Texts in Mathematics[END_REF] a semi-category C S , with set of objects Ob(C S ) = S, and such that morphisms M or(C S ) must satisfy the following properties:

• there exists an identity morphism • if a ∈ Hom(C S ), then there exists a -1 ∈ Hom(C S )

• any morphism acts on any state, which can be rephrased in the language of categories by: the semi-category is total. Thus, gathering the necessary properties of M or(C S ), we get: Proposition 2.1. Hom(C S ) is a group. This confirms the setting described in [27, section 2]. Then we have that the minimal setting for generalizing pairwise comparisons is given by actions on S by a group G, which leads to the following setting. Let I be a set of indexes and let (k, +, ., |.|)be a field with absolute value and V k a normed k-vector space. Definition 2.2. [START_REF] Koczkodaj | A Geometric Framework for the Inconsistency in Pairwise Comparisons[END_REF][START_REF] Magnot | A mathematical bridge between discretized gauge theories in quantum physics and approximate reasoning in pairwise comparisons[END_REF] Let (G, .) be a group. A pairwise comparisons matrix is a matrix A = (a i,j

) (i,j)∈I 2 such that (1) ∀(i, j) ∈ I 2 , a i,j ∈ G. (2) ∀(i, j) ∈ I 2 , a j,i = a -1 i,j . (3) a i,i = 1 G .
We note by P C I (G) the set of pairwise comparisons matrices indexed by I and with coefficients in G. When G is not abelian, there are two notions of inconsistency:

• A is covariantly consistent if and only if ∀(i, j, k) ∈ I 3 , a i,k = a i,j a j,k

• A is contravariantly consistent if and only if ∀(i, j, k) ∈ I 3 , a i,k = a j,k a i,j . Contravariant consistency appears in the geometric realization of P C I (G) via the holonomy of a connection on a simplex [START_REF] Koczkodaj | A Geometric Framework for the Inconsistency in Pairwise Comparisons[END_REF], but we give the following easy remark: Remark 2.3. Let A = (a i,j ) (i,j)∈I 2 be a contravariant PC-matrix. Then the G-matrix B = (b i,j ) (i,j)∈I 2 defined by

∀(i, j) ∈ I 2 , b i,j = a -1 i,j is a covariant PC matrix.
This shows that the two notions are dual, and we concentrate our efforts on covariant consistency in this section, that we call consistency. Definition 2.4. Let v k be a vector space equipped with semi-norms. A (non normalized, non covariant) inconsistency map is a map

ii : P C I (G) → V k such that ii(A) = 0 if A is consistent. Moreover, we say that ii is faithful if ii(A) = 0 implies that A is consistent.
We note by CP C I (G) the set of consistent PC-matrices.

After that, since V k is a vector space equipped with a semi-norm, the semi-norm will give us the "score" of inconsistency, as in the previous section. One can assume for the sake of simplicity that V k is a (normed) Euclidian space.

Algebraic properties on P C I (G)

First, we give the following easy proposition: Proposition 3.1. Any morphism of group a : G → G extends to a map ā : P C I (G) → P C I (G ) by action on the coefficients, and:

• If A ∈ P C I (G) is consistent, then ā(A) ∈ P C I (G )is consistent. • If Ker(a) = {e G }, then A ∈ CP C I (G) if and only if ā(A) ∈ CP C I (G )
We call G I the gauge group of G, following [START_REF] Koczkodaj | A Geometric Framework for the Inconsistency in Pairwise Comparisons[END_REF]. Then we get the following actions:

• a left action L :

G I × P C I (G) → P C I (G) defined, for (g i ) I ∈ G I and (a i,j ) I 2 ∈ P C I (G) by L (gi) I ((a i,j ) I 2 ) = (b i,j ) I 2 with b i,j =    1 if i = j g i a i,j if i < j a i,j g -1 j if i > j • a right action R : P C I (G) × G I → P C I (G) defined, for (g i ) I ∈ G I and (a i,j ) I 2 ∈ P C I (G) by R (gi) I ((a i,j ) I 2 ) = (b i,j ) I 2 with b i,j =    1 if i = j a i,j g j if i < j g -1 i a i,j if i > j • an adjoint action Ad (gi) I = L (gi) I • R (gi) -1 I = R (gi) -1 I • L (gi) I • a coadjoint action ((a i,j ) I 2 , (g i ) I ) → Ad (gi) -1 I ((a i,j ) I 2 ) . Theorem 3.2. [22] When I ⊂ Z, ∃(λ i ) i∈I , a i,j = λ i .λ -1 j ⇔ A is consistent.
We rephrase it the following way, extending it to any totally ordered set of indexes I: Theorem 3.3. Consistent PC-matrices are the orbits of the PC-matrix (1) I 2 with respect to the adjoint action.

Proof. Let A = (a i,j ) I 2 be a consistent PC matrix. Let i 0 ∈ I be a fixed index, and set g i = a i,i0 . Since A is consistent,

a i,j = a i,i0 a i0,j = a i,i0 a -1 j,i0 = g i .1.g -1 j
Let us give the following trivial proposition: Proposition 3.4. L and R are effective actions.

One can wonder whether L and R are free or transitive. Let us consider the following "layered cake" example:

Let λ ∈ R * + -{1}. Let us consider the matrix

A =   1 λ λ -k λ -1 1 λ λ k λ -1 1   .
Let us calculate the orbit of A with respect to the left action (with the special case

G is abelian.) Let g ∈ (R * + ) 3 . L g (A) ∈ CP C 3 (R * + ). L g (A) =   1 g 1 λ g 1 λ -k g -1 1 λ -k 1 g 2 λ g -1 1 λ k g -1 2 λ -1 1  
In this example, g 3 is not acting, so that L is not free. Let us solve

L g (A) =   1 1 1 1 1 1 1 1 1  
we get the uncompatible equations:

   g 1 = λ -1 g 2 = λ -1 g 1 = λ k So that A is not in the orbit of   1 1 1 1 1 1 1 1 1
  for the left action, and hence the left action is not transitive. However, one could wonder whether the orbits of the left action intersect CP C I . With the same example, let us try to solve "L g (A) is consistent", we get

g 1 g 2 λ 2 = g 1 λ -k which gives g 2 = λ -2-k .
This gives a one parameter family of solutions

(g 1 , g 2 ) = (t; λ -2-k ), t ∈ R * + .
Generalizing this, we give: Theorem 3.5. If I = N 3 , any orbit for the left action intersects CP C 3 (G). If card(I) > 3, there exists orbits for the left action which do not intersect CP C I (G).

Proof. Let A ∈ P C 3 (G) and g = (g 1 , g 2 , g 3 ) ∈ G 3 L g (A) =   1 g 1 a 1,2 g 1 a 1,3 a 2,1 g -1 1 1 g 2 a 2,3 a 3,1 g -1 1 a 3,2 g -1 2 1  
We want to find g in order to make L g (A) consistent. We get the following relation, among others:

g 1 a 1,2 g 2 a 2,3 = g 1 a 1,3 a 2,1 a 1,3 = a 2,3 g -1 2 which gives g 2 = a 2,1 a 1,3 a 3,2
This condition gives the consistent PC-matrix:

L g (A) =   1 g 1 a 1,2 g 1 a 1,3 a 2,1 g -1 1 1 a 2,1 a 1,3 a 3,1 g -1 1 a 3,1 a 1,2 1   .
Let us now consider

A ∈ P C 4 (G) and g = (g 1 , g 2 , g 3 , g 4 ) ∈ G 4 . Then L g (A) =       1 g 1 a 1,2 g 1 a 1,3 g 1 a 1,4 a 2,1 g -1 1 1 g 2 a 2,3 g 2 a 2,4 a 3,1 g -1 1 a 3,2 g -1 2 1 g 3 a 3,4 a 4,1 g -1 1 a 4,2 g -1 2 a 4,2 g -1 3 1       .
We then apply the procedure given for 3 × 3 PC matrices on the diagonal 3 × 3 blocks. This gives, for i ∈ {2; 3} :

g i = a i,i-1 a i-1,i+1 a i+1,i
and reporting this equality in the matrix, we get

L g (A) =       1 g 1 a 1,2 g 1 a 1,3 g 1 a 1,4 a 2,1 g -1 1 1 a 2,1 a 1,3 a 2,1 a 1,3 a 3,2 a 2,4 a 3,1 g -1 1 a 3,1 a 1,2 1 a 3,2 a 2,4 a 4,1 g -1 1 a 4,2 a 2,3 a 3,1 a 1,2 a 4,2 a 2,3 1
      .
So that, consistency now depends on the first line and the first column, and we get the relations:

g 1 a 1,2 a 2,1 a 1,3 a 3,2 a 2,4 = g 1 a 1,4 (3.1) g 1 a 1,3 a 3,2 a 2,4 = g 1 a 1,4 (3.2)
After simplifying g 1 , we gather the two lines give the same condition

(3.3) a 1,4 = a 1,3 a 3,2 a 2,4 .
This condition is not fulfilled, unless in very special cases. For an arbitrary P C I (G), with card(I) > 4, we extract a 4 × 4-PC-matrix to get the same result.

Let us now turn to other properties inconsistency maps.

Definition 3.6. Let ii be an inconsistency map. It is called:

• normalized if ∀A ∈ P C I (G), ||ii(A)|| ≤ 1. • Ad-invariant if ∀A ∈ P C I (G), ∀g ∈ G I , ii (Ad g (A)) = ii(A) • norm invariant if ||ii(.)||is Ad-invariant.
Let F I (G) be the quotient space for the Adjoint action of the gauge group on P C I (G). Next result is a classical factorization theorem: Theorem 3.7. An Ad-invariant inconsistency map ii factors in an unique way through the maps ii = f • π where -π :

P C I (G) → F I (G) is the quotient projection -f ∈ V F I (G) k .
We give also the following easy proposition: Proposition 3.8. Morphisms of groups are acting by pull-back on inconsistecy indicators. Moreover, the pull-back of a normalized (resp. Ad-invariant) inconsistency map is a a normalized (resp. Ad-invariant) inconsistency map.

According to [START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons Matrices Revisited[END_REF] and generalizing to any group G, we give now the following definition: Definition 3.9. An inconsistency indicator ii on P C I (G) is a faithful, normalized inconsistency map with values in R + such that there exists an inconsistency map ii 3 on P C 3 (G) that defines ii by the following formula

ii(A) = sup {ii 3 (B) B ⊂ A; B ∈ P C 3 (G)} .
We remark here that since ii is faithful, it is in particular (trivially) Ad-invariant on CP C I (G), but we do not require it to be Ad-invariant. Moreover, with such a definition, to show that ii is Ad-invariant, it is sufficient to show that ii 3 is Ad-invariant. However, we give the example driven by Koczkodaj's approach. This is already proved that Kii 3 generates an inconsistency indicator [START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons Matrices Revisited[END_REF] and we complete this result by the following property:

Proposition 3.10. Let n ≥ 3. Koczkodaj's inconsistency maps Kii 3 and Kii n generate is Ad-invariant inconsistency maps on P C n (R * + ). Proof. This follows from straightforward computations of the type:

λ 1 a 1,3 λ -1 3 λ 1 a 1,2 λ -1 2 λ 2 a 2,3 λ -1 3 -1 = a 1,3 a -1 2,3 a -1 1,2 .

Generalization: comparisons on a graph

We consider in this section a family of states (s i ) I such that any s i cannot be a priori compared directly with any other s j . This leads us to consider a graph Γ I linking the elements which can be compared. For example, in the previous sections, Γ I was the 1-skeleton of the simplex. For simplicity, we assume that Γ I is a connected graph, and that at most one vertex connects any two states s i and s j . We note this (oriented) vertex by < s i , s j >, and the comparison coefficient by a i,j . By the way, we get a pairwise comparisons matrix A indexed by I with "holes" (with virtual 0-coefficient) when a vertex does not exist, and for which a -1 j,i = a i,j .

Example. Let us consider the graph Γ 5 , with 5 states decribed figure 1.

A PC-matrix on Γ 5 is of the type:

      1 a 1,2 a 1,3 a 1,4 a 1,5 a -1 1,2 1 0 0 0 a -1 1,3 0 1 0 0 a -1 1,4 0 0 1 a 4,5 a -1 1,5 0 0 a -1 4,5 1
      .
4.1. Hierarchyless comparisons, "hearsay" evaluation and holonomy on a graph. In this model, the comparison between two states s i and s j can be performed by any path between s i and s j of any length. One can think about the propagation of rumours, where validation of information is based on hearsay results. With this approach, the capacity of propagation of an evaluation is not controlled. We note by

< s i1 , ..., s i k >=< s i1 , s i2 > ∨...∨ < s i k-1 , s i k >
the composition of paths along vertices. By analogy with the holonomy of a connection, we define: Definition 4.1. Let s = s i and s = s j be two states and let

H s,s = a i,i2 ...a i k-1 ,i | < s, s i2 , ..., s i k-1 , s > is a path from s to s .
We note by H s the set H s,s .

By the way, we get the following properties, usual for classical holonomy and with easy proof: Proposition 4.2.

(

) 1

4.2.

Ranking the trustworthiness of indirect comparisons. The main problem with hierarchiless comparisons of two states s and s is that paths of any length give comparison coefficients which cannot be distinguished. An indirect comparison, given by a path with 3 vertices, has the same status as a comparison involving a path with 100 vertices. This is why we need to introduce a grading on H s,s called order. This terminology will be justified by the propositions thereafter.

Definition 4.4. Let s and s be two states.

• Let γ be a path on Γ I from s to s . The length of γ, noted by l(γ), is the number of vertices of γ, and by H(γ) its holonomy. • Let h ∈ H s,s . The order of h is defined as

ord(h) = min {l(γ) |H(γ) = h } .
As a trivial consequence of the triangular equality, and as a justification of the terminology, we have:

Proposition 4.5. Let s, s a,d s three states. Let (h, h ) ∈ H s,s × H s ,s . Then ord(hh ) ≤ ord(h) + ord(h ).
Left action, right action and adjoint action of G I extend straightway to PCmatrices on Γ I setting ∀g ∈ G, g.0 = 0.g = 0.

Adapting the proof of Theorem 3.3 we get:

Theorem 4.6. Let A = (a i,j ) (i,j)∈I 2 be a PC matrix on Γ Then A is consistent if and only if there exists (λ i ) ∈ G I such that a i,j = λ -1 i λ j when a i,j = 0.

Let A be a PC matrix on Γ I . Inconsistency will be given here by the holonomy of a loop. Let us recall that a trivial holonomy of a loop < s i1 , s i2 , ...,

s i k , s i1 > implies that a i1,i k a i1,i2 ...a i k-1 ,i k -1 = 1.
This relation has to be compared with formula (1.2). The principle of ranking inconsistency with loop lengthgives the following: One can easily see that a n generalize Kii n , and Kii N is a R[[X]]-valued inconsistency map. Adapting Proposition 3.10, we get the following property: Proposition 4.8. Kii n is an Ad-invariant inconsistency map if and only if F is Ad-invariant.

Minimizing inconsistency and the necessity of topological

structures on G.

Distance matrix.

In this section, G = R * + . Setting k i,j = | log(a i,j )| we get another matrix, that we define as the distance matrix

K = (k i,j ) (i,j)∈I 2 .
Notice that, if the coefficients of this matrix satisfy the triangle inequality ∀(i, j, l) ∈ I 3 , k i,l ≤ k i,j + k j,l , we get a curvature matrix for metric spaces [START_REF] Gromov | Metric structures in Riemannian and non-Riemannian spaces[END_REF]. Due to the absolute value, we have the following: Proposition 5.1. Let K be a non zero distance matrix on ∆ n . Let N be the number of non zero coefficients in K. Then N is even and there exist 2 N/2 corresponding PC matrices.

Outline of proof. For each k i,j = 0, log(a i,j ) has 2 possible signs. Therefore, we have the following results: Proposition 5.2. Let K be the distance matrix on ∆ n associated to a consistent PC matrix A, which is assumed to be non zero. Let N be the number of coefficients k i,i+1 which are non zero. Then there exists 2 N consistent PC matrices built with the coefficients k i,i+1 , but only 2 consistent ones, A and its transposition.

Proof. The first part of the proof follows the last proposition: the sign of ln(a i,i+1 ) gives the 2 N consistent PC matrices which correspond to the coefficients k i,i+1 . However, for any coefficient k i,l , with l > i, the formula

a i,k = l-1 j=i a j,j+1
shows that there are two possible choices: ln a 0,1 = ln a 0,1 or ln a 0,1 = -ln a 0,1 , which determines the sign of the other coefficients of the transposed matrix A T . So that, PC matrices cannot be encoded as distance matrices in their own generalities when G = R * + . This suggests that the approach with G-distances suggested in [START_REF] Koczkodaj | Inconsistency indicator maps on groups for pairwise comparisons[END_REF] does not generalize PC matrices but only describes a different, quite similar tool. 5.2. Vincinities, neighbourhoods of CP C I and inconsistency maps. Let us remark that for I = N 3 , P C 3 (G) splits by various ways into

CP C 3 (G) × G.
One map which realize this one-to-one correspondence is

Φ 3 :   1 a 1,2 a 1,3 a 2,1 1 a 2,3 a 3,1 a 3,2 1   ∈ P C 3 (G) →     1 a 1,2 a 1,2 a 2,3 a 2,1 1 a 2,3 a 3,2 a 2,1 a 3,2 1   , a 1,2 a 2,3 a 3,1   ∈ CP C 3 (G) × G.
By the way, F 3 (G) can be identified with G. The same way, one can define:

Φ n : (a i,j ) (i,j)∈N 2 n ∈ P C n (G) → (b i,j ) (i,j∈N 2 n ) , a 1,2 a 2,3 a 3,1 , a 2,3 a 3,4 b 4,2 , ..., a 1,2 a 2,3 a 3,4 a 4,1 , ... ∈ CP C n (G)×G (n-1)(n-2) 2
, where the G-coefficients b i,j are defined by :

• b i,i = 1 G , • ∀i ∈ N n-1 , b i,i+1 = a i,i+1 and b i+1,1 = b -1 i,i+1 = a i+1,i , • ∀i ∈ N n-1 , ∀p ∈ N n-1-i , b i,i+1+p = a i,i+1 a i+1,i+2 ...a i+p,i+p+1 and b i+1+p,i = b -1 i,i+1+p
, which defines a consistent PC matrix. Notice that we recognize in the G (n-1)(n-2) 2

-component the coefficients present in the formulas for the generalization of Koczkodaj's inconsistency indicator given in Definition 4.7. Moreover, changing the PC matrix (a i,j ) (i,j)∈N 2 n by Adjoint action of an element (g i ) i∈Nn of the gauge group G n into (g i .a i,j .g -1 j ) (i,j)∈N 2 n we get that:

• the matrix (b i,j ) (i,j)∈N 2
n is also changed the same way into (g i .b i,j .g -1 j ) (i,j)∈N 2 n

• the coefficients a i,i+1 ...a i,i+k a i+k,i in the

G (n-1)(n-2) 2
-component are changed by classical Adjoint action of G on itself, into g i .a i,i+1 ...a i,i+k a i+k,i g -1 i . We turn now to inconsistency maps. Here the notion of inconsistency indicator is not necessary since we principally look at what happens when the value of the inconsistency map is at a neighbourhood of 0. An inconsistency map ii pulls back any topological filter in V k into a topological filter in P C n (G). One of particular interest is the filter F(V k ) of neighbourhoods of 0 in V k .

Definition 5.3. The fundamental filter of an inconsistency map ii is the filter ii * (F(V k )) .

The following proposition is straightforward: Proposition 5.4. An inconsistency map ii on P C n (G) is faithful if and only if

A∈F (V k ) ii -1 (A) = CP C n (G).
Indeed the very wide variety of such filters around CP C n (G), even if inconsistency maps ii can be, in a first approach, assumed faithful and C 0 -maps, shows that there is at this step no way to decide how an inconsistency map can be better than another. However, the notion of Ad-invariance of inconsistency maps can furnish a preferred restricted class. Gathering the results given before, we get that

F n (G) ∼ G (n-1)(n-2) 2
.

Remark that this identification is not Ad-invariant (unless G is abelian), where as F n (G) is precisely the space of the orbits of the Adjoint action of the gauge group G n . So that,one can generate faithful Ad-invariant inconsistency maps from functions f :

F n (G) → V k such that f -1 (0) is the orbit CP C n (G).
In this case, orbits are level lines of ii = f • π. Koczkodaj's inconsistency indicator is such an example.

Differential geometric methods on PC matrices

We now recall a techincal but well-known result: Theorem 6.1 (Yamabe's Theorem). [START_REF] Yamabe | On an arcwise connected subgroup of a Lie group[END_REF] Any locally compact topological subgroup of an analytic Lie group is an analytic Lie group.

Hence, from the natural topological properties derived from the previous section, we can assume with almost no loss of generality that G is a Lie group, at least for groups which have a presentation into matrix groups.

6.1. PC matrices read on a simplex. An exposition on holonomy is given in [START_REF] Kobayashi | Foundations of Differential Geometry[END_REF][START_REF] Kolar | Natural operations in differential geometry Springer[END_REF][START_REF] Kriegl | The convenient setting for global analysis Mathematical surveys and monographs[END_REF][START_REF] Magnot | q-deformed Lax equations and their differential geometric background Lambert[END_REF]. The geometry of simplexes is well addressed by [START_REF] Friedman | Survey article: an elementary illustrated introduction to simplicial sets[END_REF][START_REF] Whitney | Geometric Integration Theory[END_REF], and the notion of Lie group is described in [START_REF] Kobayashi | Foundations of Differential Geometry[END_REF][START_REF] Kriegl | The convenient setting for global analysis Mathematical surveys and monographs[END_REF]. Examples of finite dimensional Lie groups are provided by (classical) groups of (invertible) matrices, where multiplication and inversion are smooth coefficientwise. Other examples can be provided using a very general framework of differentiable manifolds, but finite dimensional Lie groups can be realized as groups of matrices.

Let (G, .) be a Lie group with Lie algebra (g, +, [., .]). The expression "Lie group" is here understood in a very general sense. This can be a finite dimensional or an infinite dimensional group, or even a Frölicher group with Lie algebra [START_REF] Magnot | q-deformed Lax equations and their differential geometric background Lambert[END_REF]. However, the beginner in the topic of Lie group is strongly advised to consider G as a matrix group, and we furnish in the appendix a short intuitive introduction of Lie groups for beginners in order to help if needed, and can skip the end of this short paragraph. The only technical requirement for the sequel is the existence of an exponential map G). This ensures the existence of the holonomy of a connection [START_REF] Magnot | Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation[END_REF]. Such a property is always fulfilled for finite dimensional groups, but not for Frölicher Lie groups. We get an example of Frölicher Lie group with no exponential map considering G = Dif f + (]0; 1[), the group of increasing diffeomorphisms of the open unit interval [START_REF] Magnot | The group of diffeomorphisms of a non compact manifold is not regular[END_REF].

exp : C ∞ ([0; 1], g) → C ∞ ([0; 1], G) solving the logarithmic equation g -1 .dg = v, where v ∈ C ∞ ([0; 1], g) and g ∈ C ∞ ([0; 1],
On a trivial principal bundle P = M × G, the horizontal lift of a path c ∈ C ∞ ([0; 1], M ) from a starting point p = (c(0), g 0 ) ∈ M × G with respect to a connection θ is the path c = (c, g) ∈ C ∞ ([0, 1], P ) such that g -1 .dg = θ(dc). If c is a loop, we have Hol g(0) c = g(0) -1 .g [START_REF] Bozóki | On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices[END_REF]. The holonomy of a loop depends on the base point (γ(0), g(0)) and is invariant under co-adjoint action. In this text, the coadjoint action is understood as an action of G on the (total) space T G = G × g in the spirit of [START_REF] Milnor | Remarks on infinite-dimensional Lie groups, Relativity, groups and topology[END_REF], which allows the same notation for the co-adjoint action of G on itself or on its Lie algebra g.

Let n ∈ N * and

∆ n = (x 0 , . . . , x n ) ∈ R n+1 | n i=0 x i = 1 ∧ (∀i ∈ {0, . . . .n}, x i ≥ 0)
be an n-simplex. This simplex can be generalized to the infinite dimension:

∆ N = (x n ) n∈N ∈ l 1 (N, R * + )| ∞ i=0 x i = 1
and

∆ Z = (x n ) n∈Z ∈ l 1 (Z, R * + )| i∈Z x i = 1 ,
where the summation over Z is done by integration with respect to the counting measure. In the sequel, ∆ will denote ∆ n , ∆ N or ∆ Z . Since ∆ is smoothly contractible, any G-principal bundle over ∆ is isomorphic to ∆×G and a G-connection 1-form on ∆ is a 1-form θ ∈ Ω 1 (∆, g), which extends to a G-covariant 1-form in Ω 1 (∆, g), with respect to the coadjoint action of G on g. We define a gauge (g i ) i∈I ∈ G I with γ i (1) = (γ i (1), g i ) where

γ i = [s 0 , s 1 ] * . . . * [s i-1 , s i ] if i > 0 and γ i = [s 0 , s -1 ] * . . . * [s i+1 , s i ] if i < 0. We set g i = Hol (s0,1 G ) γ i .
a i,j = s j .Hol (s0;e G ) γ i [s i , s j ]γ -1 j .s -1 i .
In the light of these specifications, we set

A = M at(a i,j )
and the required notion of consistency is contravariant consistency. Proposition 6.2. A is a PC matrix.

Proof. This follows from holonomy in "reverse orientation".

Let γ i,j,k = γ i * [s i , s j ] * [s j , s k ] * [s k , s i ] * γ -1 i
be the loop based on s i along the border of the oriented 2-vertex [s i , s j , s k ], where * is the composition of paths. By Equation (6.1), contravariant consistency seems to fit naturally with flatness of connections:

∀i, j, k, a i,k = a j,k .a i,j ⇔ a k,i .a j,k .a i,j = a i,i = 1 G ⇔ Hol(γ i,j,k ) = 1 G
It is necessary to examine whether every PC matrix can be expressed as a matrix of holonomies of a fixed connection. For this, we need to assume that the group G is exponential, which means that the exponential map g → G is onto. Theorem 6.5. The coadjoint map

Ad : G I × {contravariant PC matrices} → {contravariant PC matrices} (g i ) i∈I × (a j,k ) (j,k)∈I 2 → (g -1 k a j,k gj ) (j,k
)∈I 2 transforms consistent PC matrices to consistent PC matrices. Moreover, for each (g i ) i∈I ∈ G I , the map Ad (gi) i∈I is one-to-one and onto.

Let us now recall some basics on the gauge group on ∆ × G. This group is given by C ∞ (∆, G) and acts on the space of connections Ω 1 (∆, G) by the formula:

(g, θ) ∈ C ∞ (∆, G) × Ω 1 (∆, G) → g -1 dg + Ad g -1 θ.
Under these conditions, the holonomy Hol p (γ) of a loop γ transforms into Ad g -1 (Hol p (γ)) at the same basepoint p of the principal bundle ∆ × G. Theorem 6.6. If G is a compact exponential finite dimensional Lie group, then for each (g i ) i∈I ∈ G I , there exists g ∈ C ∞ (∆, G) such that g(s i ) = g i . Moreover, for any contravariant PC matrix A which is the holonomy matrix of a connection θ, then Ad (gi) i∈I (A) is the holonomy matrix of the connection g -1 dg + Ad g -1 θ.

Proof. Since G is exponential, setting cut(G) to the cut-locus of G with respect to the exponential map, we have that G -cut(G) is star-shaped. So, there exists a continuous map g : ∆ → G which restricts to ∆ -∂∆ → G -cut(G) such that g(s i ) = gi . Since G is compact (and hence finite-dimensional), this map can be chosen smooth [START_REF] Hirsch | Differential Topology[END_REF]. Once this map is constructed, the rest of the theorem follows from classical properties of the gauge group action that we have sketched before.

6.3.

When G is a free abelian Lie group. Assume that G = (R * )

J where J has any cardinality, finite or infinite. In this case,

Hol(< s i , s j >) = e 1 0 θ(dsi,j )
where ds is the unit vector of the normalized length parametrization of [s i , s j ]. Thus,

Hol(γ i,j,k ) = 1 G ⇔ 1 0 θ(ds i,j ) + 1 0 θ(ds j,k ) + 1 0 θ(ds k,i ) = 0 (6.3)
The connection θ is flat now and it reads as dθ = 0 which is equivalent to θ = df, where f ∈ C ∞ (∆, R J ) (because H 1 (∆, R) = 0). With this function f , setting e f (si) = λ i , we recover the "basic consistency condition" of Theorem 3.2. In the spirit of Whitney's simplicial approximation [START_REF] Whitney | Geometric Integration Theory[END_REF], we assume that G = R * and g = R for simplicity, and our computations will extend to R J componentwise. Let us construct an affine function f . This function is uniquely determined by its values f (s i ), for i ∈ {0, . . . , n} and we get the system: (6.4)

       f (s 0 ) -f (s 1 ) = -ln(a 0,1 ) f (s 1 ) -f (s 2 ) = -ln(a 1,2 ) (...) f (s n-1 ) -f (s n ) = -ln(a n-1,n )
which is a n-system with (n + 1) variables. Since θ = df, we can normalize it, assuming e.g. f (s 0 ) = 0, and the system gets a unique solution, and hence a unique affine function f and an unique connection θ = df. Now, setting λ i = e f (si) , we recover the construction given in the proof of theorem 3.2 for this particular choice of group G.

6.4. Remarks on methods for minimizing inconcistency. Let us fix the set P C n (G) and an inconsistency indicator ii. We assume here that the chosen inconsistency indicator ii is C 0 , R + -valued and of C k -regularity off the set CP C n (G), with k large enough. Passing from G = R * + to a general Lie group G, most techniques used in the existing scientific literature on pairwise comparisons cannot be applied straightway because they are based on three properties of R * + which are still not valid on a Lie group G, namely:

(1) R * + is contractible; (2) ∀x ∈ R * + , ∃!y ∈ R * + , y n = x; (3) R *
+ is abelian. However, the inconsistency indicator ii appears like a functional which has to be minimized. This kind of problem has been approached by many ways in optimization techniques, where enough regularity of the functional is needed. We leave these questions for future works, because the framework of application may impose to choose an optimization scheme or another. We also have to highlight that this problem of minimization for functional has a pending problem in the field of mathematical physics. In particular, when ii is Koczkodaj's inconsistency indicator, minimizing ii(A) = sup{ii 3 (B) | B ⊂ A} is similar to minimizing the holonomy along boundaries of 2-simplexes. This approach leads to consider Yang-Mills type functionals along the lines of [START_REF] Magnot | A mathematical bridge between discretized gauge theories in quantum physics and approximate reasoning in pairwise comparisons[END_REF][START_REF] Magnot | Remarks on a new possible discretization scheme for gauge theories[END_REF], which will be studied extensively in a future work.

Uncertainty and precision of measurements in pairwise comparisons

We now turn to probabilistic aspects. For this we must assume, a priori, that there exists a bi-invariant Radon measure λ on the topological group G, which is also called Haar measure. For such a measure, for any integrable function f : G → R, we have the bi-invariance properties of integration with respect to λ : x dx.

∀g ∈ G, G f (x)dλ(x) = G f (g.x)dλ(x) = G f (x.g)dλ(x).
(3) If G is compact, the Haar measure has finite volume and can be normalized into a probability measure.

On one hand, considering only the Haar measure would mean that the measurements are totally random, without any consideration on the real situation under evaluation. On the other hand, exact evaluations belong to an idealist picture, and errors can occur from many different ways: measurements, expert unability among others. Moreover, error perception is limited to a fixed precision. These are the reasons why a measurement is better modelized by a probability distribution which "concentrates" around its expectation value which is chosen as th exact value of the measurement. Among the most known examples, Gaussian measures play a very important role. On R, a Gaussian probability measure γ such that E(γ) = 0 reads as

(7.1) γ(A) = 1 √ 2πσ 2 A e -x 2 2σ 2 dλ.
When σ → 0 + , γ converges, for the vague topology of probability measures, to the Dirac measure δ 0 defined by δ 0 (A) = 1 if 0 ∈ A, and δ 0 (A) = 0 if 0 / ∈ A. In this picture, 0 reads as the exact measure, and µ(A) reflects with which probability the measurement obtained is in the set A ⊂ R.

We propose here two ways of generalization of this picture to pairwise comparisons matrices. In these two approaches, we assume that each coefficient a i,j is a random variable.

• On one hand, let us consider a measure γ i,j on G which models, for any measurable set U ⊂ G, the probability to get a i,j ∈ U after "expert evaluation".

Identifying P C n (G) with G n(n-1) 2
we get a product measure ⊗ 1≤i<j≤n γ i,j on P C n (G). Thus for any inconsistency map ii : P C n (G) → R + , assuming that inconsistency of A ∈ P C n (G) is acceptable if ii(A) < for a fixed value > 0, the probability for the acceptable inconsistency is given by

⊗ 1≤i<j≤n γ i,j ({A ∈ P C n (G) | ii(A) ≤ }) .
• On the other hand, let us assume that the chosen inconsistency map ii reflects the global perception of the inconsistency of the system of an extraobserver, whose natural tendency will lead him to minimize inconsistency, ideally at the level ii < .Then we can propose a Feyn-man-Kac type formula by introducing the (normalized) measure µ ii of density e -ii(a) with respect to the Haar probability measure λ on G n(n-1) 2

∼ P C n (G) given by the formula:

µ(A) = 1 Z A e -ii(x) dλ(x) with Z = P Cn(G)
e -ii(x) dλ(x)!;

: P C I (GL n (R)) → R + A = (a i,j ) i∈I → ii R * + (|det(a i,j )|) (i,j)∈I 2
, defined as a composed map

ii det = ii R * + • |det(.)
|, is a non-faithful, Ad-invariant inconsistency operator on P C I (GL n (R)).

Proof. The only non trivial part is non-faithfulness. For this, let us give a counterexample. Let A ∈ P C 3 (GL 2 (R)) defined by a 1,2 = a 2,3 = -a 1,3 = I 2 . Then a 1,2 a 2,3 = a 1,3 where as

|det(A)| =   1 1 1 1 1 1 1 1 1   ∈ CP C 3 (R * + ), thus ii det (A) = 0.
But this class of inconsistency maps is not the only one of interest, even if det generates Hom(GL n (R), R * ) in the category of groups. This toy example shows how situations expressed by a (more) complex group such as GL n (R) cannot be reduced straightway (one would say naively) to a more simple abelian group such as R * + . 8.2. Error in cartography and in tunnel building. Let us now describe two situations where non-abelian groups rise naturally in a description of errors by the pairwise comparisons method. For these two examples, the group under consideration is a group of orientation preserving, isometric affine transformations Is n of a (finite dimensional) affine space R n .

Example: cartography in a forest One of the main features in cartography, or during the recovery of an exit path, in a forest (here in a flat land) is the lack of external point where to get a precise indication on the actual position of the observer. By the way, moves and direction changes can only be appreciated by self-evaluation, which is subject to numerous, non-compensative errors in the appreciation of the positions during the path. More precisely, (1) moves along a straight line can be evaluated by a translation, i.e. a vector in R 2 (2) changes of direction can be evaluated as rotations, centrered at the position of the observer.

Gathering theses two aspects, a path in teh forest can be assimilated to succession of moves transcribed by elements a 1,2 , ...a n-1,n of the group generated by planar rotations and translations. This group is the group of orientation preserving, isometric affine maps in R 2 . Thus if one can evaluate the n -th position with respect to the initial one, this gives another element a 1,n . Due to successive errors, very often a 1,n = a 1,2 a 2,3 ...a n-1,n .

This is exactly a situation of inconsistency where coefficients are in the non-abelian group Is 2 .

Example: error in tunnel building The situation is the same in tunnel building, where the surveyors need to indicate, at each step of perforation of a tunnel, in which direction one has to correct the next perforation step underground. Each tunnel starting from each side must meet exactly at the end of the process. For the same reasons as in previous example, the (non-abelian) group under consideration here is Is 3 . Currently, the admissible error is in the range of 1 cm per 100 m of tunnel. This error is admissible inconsistency.

8.3.

Perspective in image processing. Let [abcd] be a 3 simplex (tetrahedron).

Let ω ∈ R 3 \[abcd], and let P ω be a (projection) plan, such that ω / ∈ P ω . In projective perspective, the projection of x ∈ R 3 -{ω}, is x 0 ∈ P ω such that {x 0 } = (ω, x) ∩ P ω (if it exists).

Let us recall that projections exist in a "generic" way, that is, x 0 exists unless (x, ω) and P ω are parallel. We also assume that ω is "far enough" (at the "optical infinity") so that, in first approximation, projections can be asssimilated to affine projections on P ω .

Let us consider for simplicity, first, a tetrahedron [abcd] ⊂ R 3 , and let a 0 , b 0 , c 0 and d 0 the corresponding projections with respect to ω. Any 2-simplex [xyz] of [abcd] projects to a 2-simplex [x 0 y 0 z 0 ] in P ω . For another choice (ω , P ω ), we get other projections [x 0 y 0 z 0 ] of [xyz]. Now, identifying P ω and P ω , with the standard plan R 2 by an arbitrary choice of coordinates, we give the numbers 1,2,3 and 4 to the faces resp. [abc], [abd], [acd] and [bcd] and, in R 2 , define a i,j as the unique affine map R 2 → R 2 which transforms the i-th face to the j-th face for ω-the projection, and a i,j the corresponding coefficient for the ω -projection. Let us now consider the unique affine maps λ i which transforms the i -th face form the ω-projection to the ω -projection. We then have that a i,j = λ -1 j a i,j λ i for any index, and the PC-matrices (a i,j ) and (a i,j ) are consistent, but the tetrahedron is flat if and only if

λ 1 = λ 2 = λ 3 = λ 4 = λ.
Thus, this leads to the following definition: Definition 8.3. Let G be the group of affine bijections of R 2 . We call perspective matrix of [abcd] with respect to (ω, P ω ) and (ω , P ω ) the matrix (b i,j ) ∈ P C 4 (G) defined by b i,j = λ i a i,j λ -1 i , which is defined up to Ad-action.

Then we get the trivial proposition, passing through the consistency in P C 4 (G) : Proposition 8.4. A matric (b i,j ) ∈ P C n (G) encodes a triangulated simplicial complex in R 3 if and only if it is in CP C n (G).

Thus perspective of a 3-simplex can be encoded into a CP C 4 (G) matrix. Let us now have a brief look at the situation of a more complex solid that we assume triangulated, i.e. composed by a family of simplexes glued together along their faces.

Figure 1 .

 1 Figure 1. The graph Γ 5

Definition 4 . 7 .

 47 Let F : G → R + be a map such that I(1) = 0. Let s be a basepoint on Γ I . The ranked Koczkodaj's inconsistency map associated to F the mapKii N = n∈N a n X nwhere a n = sup {F (H(γ)) |γ is a loop at s and l(γ) = n} .

Figure 2 .

 2 Figure 2. holonomy of a loop c(t)

  Let us recall that, for two paths c and c such that c * c exists (i.e. c(1) = c (0)), if p = (c(0), e G ), p = (c (0), e G ) and h = Hol p c, we have: (6.1) Hol p (c * c ) = Hol p (c).(h -1 Hol p (c )h) = Hol p (c ).Hol p (c). Let (6.2)

For example, ( 1 )

 1 for n ∈ N * , (classical) Riemannian integration of functions R n → R envolve the Haar measure of the abelian group (R n , +) called Lebesgue measure. (2) the multiplicative group (R * + , .) has a Haar measure different from the Lebesgue measure on (R, +), because the group law is different. The exponential map exp : R → R * + is an isomorphism of topological groups. Setting λ and λ respectively the Lebsegue measure on Rand the Haar measure on R * + , if g : R * + → R is a C 0 -function and f = g • exp, we have R

Figure 3 .

 3 Figure 3. Two projections of [abcd]

  Let s be a state, then H s is a subgroup of G. We call it holonomy group at s.(2) Let s and s be two states. Then H s and H s are conjugate subgroups of G.(3)H si,sj = a i,j .H sj = H si .a i,j .Example. With the graph Γ 5 of figure1, H 1 is the subgroup of G generated by a 1,4 a 4,5 a 5,1 .

Definition 4.3. The PC matrix A on the graph Γ I is consistent if and only if there exists a state s such that H s = {1}.

8 .

 8 Examples 8.1. GL n -comparisons. Let S = R ∞ be the inductive limit of the family {R n ; n ∈ N * } such that the inclusions R n ⊂ R n+1 is the canonical inclusion with respect to the first coordinates. Here, S is an object of the category of vector spaces. With this setting, we get G = GL ∞ (R), which is the inductive limit of the family {GL n (R); n ∈ N * }. If I is a finite set of indexes (e.g. I = N k for some k ∈ N * ), sup i∈I dim(s i ) < +∞ and setting n = sup i∈I dim(s i ), we work with the restricted setting S = R n and G = GL n (R).Remark 8.1. Even if there exists an index i ∈ I such that n i < n, we have to consider the inclusion R ni ⊂ R n because there is no linear isomorphism from R ni to R n (by the theorem of dimension). Proposition 8.2. Let ii R * + be a (R + -valued) inconsistency map on P C I (R * + ).

	With this construction, we get a first family of inconsistency maps. The deter-
	minant map
	det : GL n (R) → R *
	is a group morphism.
	Then the map
	ii det

Proposition 6.3. If G is exponential, the map Ω 1 (∆, g) → { PC matrices } θ → the holonomy matrix is onto.

Proof. Let A = (a i,j ) (i,j)∈I 2 be a PC matrix. Let us build a connection 1-form θ ∈ Ω 1 (∆, g) such as (6.2). For this, before constructing our connection, we fix the gauge (g ii∈I ) ∈ G I by

and by

Once the gauge is fixed, we begin by dealing with each 1-vertex and use gauge covariance to extend the 1-form in Ω 1 (∆, g) constructed to a (G-equivariant) connection 1-form on ∆ × G. Firstly, by fixing indexes i < j, which holds in particular for j = i + 1, we choose v i,j ∈ g such that exp(v i,j ) = a i,j . Needless to say, the condition v j,i = -v i,j is consistent with a -1 i,j = a j,i . The group {exp(tv i,j )|t ∈ R} is an abelian subgroup of G. For this reason, formulas for holonomy on an abelian group can be used to specify a function f i,j : [0, 1] → R + .v i,j , with support in [1/3; 2/3], and such that 1 0 f i,j (s)ds = v i on the length-parametrized edge [s i , s j ]. Finding such a function is possible, and extending the G-equivariant 1-form f i,j d s on [s i , s j ] × G to a G-equivariant 1-form θ i,j on ∆ × G which is null off V i,j × G, where V i,j is a tubular neighborhood of radius > 0 of supp(f i,j ), is also possible.

Secondly, we repeat this procedure for each couple of indexes (i, j) such that i < j, and choose small enough in order to have non intersecting supports supp(θ i,j ), for example = 1/6. By setting

we get a connection θ whose holonomy matrix is given by A.

Let us provide a geometric criterion for consistency. Proposition 6.4. If the connection θ is flat, A is a contravariant consistent PC matrix.

Proof. θ is flat if and only if its curvature is null. This implies that the Lie algebra of the holonomy group is null, and since each 2-vertex [s i , s j , s k ] is contractible, the holonomy group is trivial. 6.2. The gauge group acting on the space of contravariant PC matrices. The gauge group is defined before, and we give the following easy result which justifies the terminology:


s.maljean-dubois@univ-amu.fr

  Bilan et perspectives de l'Accord de Paris(COP 21). Regards croisés, sous la direction de Marta Torre-Schaub, juriste, directrice de recherche au CNRS, constitue la publication des actes d'une journée d'études organisée le 31 mai 2016 à Paris. Les regards croisés sont ceux de différentes disciplines : en plus des contributions écrites pour la plupart par des juristes publicistes et privatistes, internistes et internationalistes, l'ouvrage comprend les points de vue d'une philosophe (Catherine Larrère) et d'un scientifique (Jean Jouzel). L'objectif de cet ouvrage est de dresser un premier bilan presque « à chaud » de l'Accord de Paris et de réfléchir aux perspectives qu'il ouvre. La démarche est bienvenue tant cet Accord ressemble à un « objet non identifié qui vient de tomber du ciel » (Yann Aguila, conclusions). Les auteurs s'attachent à montrer le changement de paradigme dans lequel s'inscrit l'adoption de l'Accord de Paris lors de la COP 21 à la fin de l'année 2015. Pour Mireille Delmas-Marty, qui livre un très bel avant-propos, il marque le passage d'un humanisme juridique anthropocentré et individualiste, inspiré par la philosophie des droits de l'homme, à un « humanisme d'interdépendance », reflétant la vision d'une Terre-mère au sein de laquelle s'imposent ou doivent s'imposer des relations plus fraternelles. Il annoncerait « moins la disparition de toute souveraineté que la transformation d'une souveraineté solitaire en souveraineté solidaire » (p. 4). L'Accord de Paris témoigne en effet d'une telle aspiration, mais les États n'ont pas abandonné pour autant la défense de leurs intérêts propres. Ils restent pour beaucoup dans une approche réaliste du droit international. Pour preuve, le lendemain de la journée d'études à la Sorbonne, le président Donald Trump annonçait vouloir sortir d'un accord contredisant les intérêts américains. Cette décision n'a pas produit d'effet domino, mais beaucoup d'États ont une position encore ambivalente, que masque souvent mal un discours très ouvert et favorable. L'ouvrage rend bien compte de ces tiraillements et plus largement des incertitudes sur l'effectivité de l'Accord. S'appuyant sur les rapports du GIEC qui ont nourri les négociations internationales (Jean Jouzel), l'Accord de Paris est relativement ambitieux dans ses objectifs de limitation du réchauffement planétaire. Il est l'est moins dans les moyens de mise en oeuvre puisqu'il repose de ce point de vue sur des contributions nationales nationalement déterminées, sans droit de regard des autres États ou des institutions nationales. Pour l'instant, force est de constater que ces contributions, toutes additionnées, ne permettent pas d'atteindre les objectifs posés par l'Accord. Mais ce dernier est aussi le résultat d'un pari : que cette démarche souple et anticipative participe à, accompagne et encourage une nouvelle dynamique à tous les niveaux, du global au local, et chez tous les acteurs, y compris non étatiques tels que les entreprises ou infra-étatiques (États fédérés, collectivités locales, villes…). Il tente ainsi, dans une démarche originale, de percer l'écran étatique. Le livre souligne le rôle essentiel de ces acteurs dans la concrétisation des objectifs et la mise en oeuvre de l'Accord de Paris (voir notamment la contribution de François Guy Trébulle). La raison d'être de l'Accord est sans doute là. Parviendra-t-il, par un développement « séquentiel et progressif » (Marianne Moliner-Dubost), à susciter une dynamique telle qu'elle conduira les États à revoir à la hausse leurs objectifs nationaux ? L'Accord tente de « bâtir le futur » (Laurence Boisson de Chazournes), mais c'est une course contre la montre qui est engagée, comme le montrent les emballements climatiques auxquels nous assistons. S'agissant du contrôle de la mise en oeuvre, l'Accord est en retrait par rapport à son prédécesseur, le protocole de Kyoto, qui créait un mécanisme beaucoup plus complet et intrusif, pouvant déboucher sur de véritables sanctions. Dans ce « net recul de la contrainte » (Marianne Moliner-Dubost), on peut sans doute voir une faiblesse. Mais là n'est pas l'esprit de l'Accord de Paris, dont le caractère incitatif et souple est également confirmé à cette aune. Cette « faiblesse » relative pourrait d'ailleurs être partiellement compensée par le développement de contentieux climatiques à l'échelle nationale, s'appuyant largement sur la coopération internationale et ses fruits (Marta Torre-Schaub). Il y a là une revendication croissante de la société civile (Agnès Michelot). Les ponts qui s'établissent entre le régime climat et le droit international des droits de l'homme sont de ce point de vue prometteurs, même si leur portée demeure encore limitée (Christel Cournil). C'est donc peut-être au niveau national que se concrétisera, au moins en partie, une justice climatique dont la reconnaissance dans l'Accord lui-même est, selon Catherine Larrère, « aussi faible que possible et même assez cynique » (p. 25). Mais la philosophe note aussi que « ce qui permet aux promesses de se réaliser, c'est la confiance qu'on a en elles, et la mobilisation pour obliger ceux qui les ont formulées à les tenir » (p. 30). Il ne faut sans doute pas oublier le rôle de cette mobilisation dans l'enjeu de concrétisation de la prophétie autoréalisatrice que doit devenir, selon ses initiateurs, l'Accord de Paris. Ici, l'ouvrage montre bien que ce texte peut donner lieu à deux lectures. Le pessimiste déplore son caractère insuffisamment contraignant et l'inadéquation entre des objectifs globaux ambitieux et des contributions nationales qui le sont moins. L'optimiste y voit plus un point de départ qu'un aboutissement. Il veut croire que cet Accord tourné vers l'avenir participera d'une « dynamique vertueuse, reflet d'un nouveau paradigme » (Yann Aguila). La COP 21 a permis à tout le monde de monter dans le bateau. Mais à l'heure où certains, de poids, annoncent qu'ils souhaitent quitter le navire, il est bien difficile de savoir où ce navire conduira l'humanité. Avec la publication du rapport du GIEC sur les conséquences d'un réchauffement supérieur à 1,5 °C, le dialogue politique sur le relèvement du niveau d'ambition (dit « Dialogue de Talanoa »), l'adoption du manuel d'application de l'Accord au menu de la COP 24 et un probable One Planet Summit n° 2, 2018 sera une année déterminante de ce point de vue. Elle sera même un « crash test » pour la coopération internationale sur le climat. L'objectif de M. Torre-Schaub est largement atteint. Le lecteur referme cet ouvrage relativement court et accessible, y compris pour un non-spécialiste, avec une vision plus claire des enjeux. Il comprend enfin pourquoi, avec tout autant d'arguments convaincants d'un côté comme de l'autre, certains voient dans cet Accord un succès et d'autres un échec. Il prend la mesure de l'ambition des quelques pages sur lesquelles a été couché l'Accord de Paris : rien de moins qu'impulser la transition de nos sociétés vers la décarbonation et maintenir ainsi notre biosphère dans un espace de fonctionnement sécurisé. Si l'Accord de Paris représente d'ores et déjà un grand succès diplomatique, nul ne peut dire encore si ce dernier se conjuguera avec un succès environnemental.Sandrine Maljean-DuboisAix Marseille Univ, Université de Toulon, Univ Pau & Pays Adour,(CNRS, UMR7318 DICE, France)