Pierre Clavier
email: clavier@math.uni-potsdam.de

L I Guo
email: liguo@rutgers.edu

AND BIN ZHANG Sylvie Paycha
email: paycha@math.uni-potsdam.de

AN ALGEBRAIC FORMULATION OF THE LOCALITY PRINCIPLE IN RENORMALISATION

Keywords: Mathematics Subject Classification. 08A55, 16T99, 81T15, 32A20, 52B20 locality, renormalisation, algebraic Birkhoff factorisation, partial algebra, Hopf algebra, Rota-Baxter algebra, multivariate meromorphic functions, lattice cones

We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on cones, we renormalise the exponential generating function which sums over the lattice points in convex cones. For a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate functions.

1. Introduction 1.1. Locality in quantum field theory. Locality is a widespread notion in mathematics and physics. In physics, the principle of locality states that an object is only directly influenced by its immediate surroundings. A theory which includes the principle of locality is said to be a "local theory". Various notions of locality are also used in analysis (local operators, local Dirichlet forms), geometry (geometric localisation, locality in index theory), algebra (localised rings) and number theory (local fields).

In classical field theory, for a classical action A(f ) = B(f, f ) given by a bilinear form

B : C ∞ c (U, C k ) × C ∞ c (U, C k ) →
C on an open subset U ⊂ R n , the locality principle translates to for any

f 1 , f 2 ∈ C ∞ c (U, C k ) if Supp(f 1 )∩ Supp(f 2 ) = ∅, then B(f 1 , f 2 ) = 0.
We interpret the binary relation ⊤ defined by [START_REF] Apostol | Modular Functions and Dirichlet Series in Number Theory[END_REF] f 1 ⊤f 2 ⇐⇒ Supp(f 1 ) ∩ Supp(f 2 ) = ∅ on pairs {f 1 , f 2 } as an independence relation on C ∞ c (U, C k ). In quantum field theory, the locality principle governs the construction of consistent subtraction (of divergences) algorithms that preserve locality during the renormalisation process. Subtracting divergences may be interpreted as resulting from the addition to the effective action of new properly chosen terms (known as counterterms) that are local polynomials in the fields and their derivatives. A systematic algorithm to subtract divergent momentum space integrals while preserving the fundamental postulates of relativistic quantum field theory including locality was proposed by Bogoliubov, Parasiuk, Hepp and Zimmermann (abbreviated BPHZ renormalisation, and based on the so-called forest formula) [START_REF] Bogoliubov | Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder[END_REF][START_REF] Hepp | Proof of the Bogolyubov-Parasiuk theorem on renormalization[END_REF][START_REF] Brouder | Noncommutative version of Borcherds approach to quantum field theory[END_REF]. More recently, Connes and Kreimer [START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF] gave an interpretation of this algorithm by means of a coproduct which enables to build -using dimensional regularisation -a renormalised map via its algebraic Birkhoff factorisation, regarded as an algebra homomorphism from the Hopf algebra of Feynman graphs to the Rota-Baxter algebra of meromorphic functions in one variable. Alternatively, following Speer [20], one can use analytic regularisation, which gives rise to a map on graphs with values in multivariate meromorphic functions. 1 In that case, locality is reflected in the fact that this map preserves locality (what we call a locality map in Definition 2.4).

Separation of supports, which in QFT reflects independence of events, also arises in the early algebraic study of locality, in terms of locality ideals, initiated by the work of H.-J. Borchers [2,[START_REF] Borcherds | Renormalisation and quantum field theory[END_REF]. Here a locality ideal is defined to be the two-sided ideal generated by commutators of test functions with space-like separated supports. Its importance comes from the fact that quantum fields satisfying the requirement of local commutativity can be regarded as Hilbert space representations of the tensor algebra annihilating the locality ideal. See [3,4] for the recent progresses initiated by R. E. Borcherds.

1.2. Locality in algebraic Birkhoff factorisation. In this paper we take an algebraic approach to investigate how locality is preserved in the renormalisation process, and choose to work in the framework of algebraic Birkhoff factorisation à la Connes and 1 In the Epstein-Glaser formalism, an analytic regularisation à la Speer yields Feynman amplitudes obeying amongst other axioms, a factorisation condition reflecting the locality principle [START_REF] Dang | Complex powers of analytic functions and meromorphic renormalization in QFT[END_REF]Theorem 10.1].

Kreimer [START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF]. Our starting point is to view locality as a symmetric binary relation comprising all pairs of independent events as in Eq. [START_REF] Apostol | Modular Functions and Dirichlet Series in Number Theory[END_REF]. Our main task is to explore the structures which are compatible with and preserve the locality, thus providing a mathematical formulation of the locality principle.

To make our point more precise, let us briefly recall the algebraic Birkhoff factorisation in the approach of Connes and Kreimer.

Theorem 1.1. (Algebraic Birkhoff factorisation, Hopf algebra version) Let H be a connected Hopf algebra and let (A, P ) be a commutative Rota-Baxter algebra of weight -1 with an idempotent Rota-Baxter operator P . Any algebra homomorphism φ : H → A factors uniquely as the convolution product

(2) φ = φ ⋆(-1) - ⋆ φ +
of algebra homomorphisms φ -: H → K + P (A) and φ + : H → K + (Id -P )(A).

As an instance of physics applications, H is the Connes-Kreimer Hopf algebra of Feynman diagrams, A is the Rota-Baxter algebra C[ε -1 , ε]] of Laurent series and φ is the regularisation map sending a Feynman diagram to the (dimensional) regularisation of the corresponding Feynman integral with Laurent expansion in A.

We can reformulate the theorem as follows: for a multiplicative regularisation map φ, the renormalised map φ + is also multiplicative. Thus renormalisation preserves multiplicativity, a property which is the driving thread underlying the Hopf algebra method introduced by Connes and Kreimer. Also, it is essential in the applications of the algebraic Birkhoff factorisation in mathematics, specifically when renormalising multiple zeta values [10, [START_REF] Guo | Renormalization of multiple zeta values[END_REF][START_REF] Manchon | Nested sums of symbols and renormalised multiple zeta values[END_REF] while preserving their shuffle product and quasi-shuffle product.

In a recent study [15] of the renormalisation of conical zeta values and Euler-Maclaurin formula on lattice cones, the algebraic Birkhoff factorisation was generalised to weaken the Hopf algebra condition of H to a connected coalgebra as well as the Rota-Baxter algebra condition of A to allow for an algebra with a decomposition A = A 1 ⊕ A 2 where only A 1 is a required to be a subalgebra of A.

Theorem 1.2. (Algebraic Birkhoff factorisation, coalgebra version) [15, Theorem 4.4] Let H be a connected coalgebra with coaugmentation H 0 = K J and let A be a commutative (unital) algebra with an idempotent linear operator P on A such that ker P is a subalgebra of A. Any linear map φ : H → A with φ(J) = 1 A factors uniquely as the convolution product

(3) φ = φ ⋆(-1) - ⋆ φ + of linear maps φ -: H → K + P (A) and φ + : H → K + (Id -P )(A) with φ ± (J) = 1 A .
As our motivation and first application of this generalised algebraic Birkhoff factorisation, H is taken to be the connected coalgebra QC, the linear span over Q of the set C of lattice cones, with the transverse coproduct, A is the algebra M Q of multivariant meromorphic functions with linear poles and rational coefficients, equipped with the direct sum M Q = M Q+ ⊕ M Q-where M Q+ is the algebra of holomorphic functions and M Q-is the space of polar germs. See Section 2.2 for details. The linear map φ is (4)

S : QC → M Q with S(C, Λ C )(z) = n∈C o ∩Λ C e n,z
which corresponds to the exponential generating function summing over the lattice points in a lattice cone (C, Λ C ). Then a natural locality issue is whenever S(C, Λ C ) and S(D, Λ D ) are orthogonal under the locality relation on M Q induced from the Q-Euclidean space, to ask whether S + (C, Λ C ) and S + (D, Λ D ) also orthogonal. Another version of the question is, if (C, Λ C ) and (D, Λ D ) are orthogonal, whether S + (C, Λ C ) and S + (D, Λ D ) are orthogonal. We reformulate this first question relative to the locality principle in renormalisation in more general terms as follows:

Problem 1.3. (Locality Conservation Principle in Renormalisation) Consider a connected coalgebra H and a commutative algebra A with a linear map P as in Theorem 1.2. Let ⊤ H ⊆ H × H and ⊤ A ⊆ A × A be relations on H and A respectively. Let φ : H → A be a linear map compatible with the relations in the sense that (φ × φ)(⊤ H ) ⊆ ⊤ A . Determine the conditions under which the renormalised map φ + is also compatible with the relations.

The first main goal of this paper is to provide a solution to this problem as a consequence of the locality generalisation (Theorem 4.10) of the algebraic Birkhoff factorisation in Theorem 1.2.

Note that the lack of multiplication in the coalgebra H means that φ and φ + are only linear maps, leaving out the other algebraic structures. However, as we will see next, it is precisely the interaction of the binary relations with all the existing and potential algebraic structures involved, that makes the locality principle work. This interaction of binary relations with algebraic structures leads us to partially defined binary operations, which we dubbed locality structures, throughout the whole hierarchy of algebraic structures from locality set up to locality algebra and locality coalgebra, then further to locality Rota-Baxter algebra and locality Hopf algebra.

1.3. Locality and partially defined operations. In mathematics one often encounters multiplications which are meaningful only partially even if they might be everywhere defined. This phenomenon was long known in number theory where certain functions are multiplicative only for coprime positive integers, for instance Euler's totient function φ(n) counting the number of integers modulo n ≥ 1 which are relatively prime to n and the Ramanujan tau function τ (n) in modular forms. In fact, such phenomena have become so prevalent that such a restricted multiplicative function is simply called a multiplicative function in number theory [START_REF] Apostol | Modular Functions and Dirichlet Series in Number Theory[END_REF]. Such operations with restrictions can be viewed in the general framework of partial algebras in universal algebra [12] (see Footnote 2).

An example of special importance for us is that of lattice cones. Even though the product given by the Minkowski sum is defined for any two convex cones, and can be extended to any two lattice cones, compatibilities with either the coalgebra structure or the regularisation maps φ : H → A, such as the exponential sum, can be expected only when the cones are orthogonal. See later sections for details on this (Propositions 3.7 and 5.2) and other instances.

This leads to another natural question: whether, in the absence of a fully defined multiplicativity that is preserved by renormalisation, as in the classical algebraic Birkhoff factorisation in Theorem 1.

(u, v)) = m A (φ(u), φ(v))
for (u, v) ∈ ⊤ H . Determine a condition under which the renormalised map φ + is also partially multiplicative.

At this point it is worthwhile to observe the mutual effects of the interplay between locality relations and the partial algebraic structures mentioned above. In one direction it allows us to pass the locality of φ represented by the relations onto the corresponding renormalised map φ + , thus giving a solution of the Locality Conservation Principle in Problem 1.3. This is achieved in Theorem 4.10. In the other direction this interplay allows us to transmit the partial multiplicativity of φ onto φ + , thus giving a solution of the Locality Product Conservation Principle in Problem 1.4. This is achieved in Theorem 5.8.

Of particular interest to us is the exponential generating sum S : QC → M Q mentioned above. In spite of the fact that the space QC of lattice cones is a genuine algebra when equipped with the extended Minkowski product, the product is not compatible with the transverse coproduct, so we do not have a bialgebra. Likewise, in the decomposition

M Q = M Q,+ ⊕ M Q,-,
even though the summand M Q,+ , the space of holomorphic germs, is a subalgebra, the summand M Q,-, the space of polar germs, is not. Hence this decomposition does not give a Rota-Baxter algebra. Furthermore, the linear map S does not send the Minkowski product on QC to the function multiplication in M Q . However if one considers only orthogonal pairs of lattice cones and suitable orthogonal relation of meromorphic germs, all these structures can be recovered in the form of locality structures. In fact QC is not only a locality bialgebra, it is a connected locality Hopf algebra. Moreover, M Q,-is not only a locality subalgebra, it is a locality ideal, showing that the projection π + : M Q → M Q,+ onto M Q,+ along M Q-is a locality algebra homomorphism. Consequently, the full algebraic Birkhoff factorisation can be recovered on the locality level, from which it then follows that the renormalised map φ + is a locality algebra homomorphism. Further the locality ideal property of M Q,-implies that its convolution inverse φ ⋆-1 + is φ composed with the projection of π + (see Eq. ( 37)). This applies whenever the regularisation map φ is a locality algebra homomorphism and takes values in M Q . In this situation a recursive formula for φ -in terms of the projections π + and Idπ + is given in (29), which is reminiscent of the forest formula of the renormalisation of Feynman graphs in the context of Quantum Field Theory.

1.4. Outline of the paper. We next give a summary of the locality construction as an outline of the paper.

We consider vector spaces H and A, each equipped with a suitable symmetric binary relation, and a linear map φ : H → A preserving the relations. In order to pass this property of φ onto the corresponding renormalised map φ + via the algebraic Birkhoff factorisation for a suitably enriched H, A and φ, we need to make the relations compatible with all the algebraic structures involved in the algebraic Birkhoff factorisation, including a Hopf algebra or bialgebra structure on H, a Rota-Baxter algebra or algebra structure on A, and the corresponding structures on φ.

Throughout the paper we use the space of convex lattice cones, the space of meromorphic functions and the exponential generating sum in Eq. (4) between the two spaces as the primary examples of the algebraic constructions, and of our main theorems on locality. Further applications will be given in subsequent work, such as locality of branched zeta values [7]. The locality relation relates to Weinstein's "congeniality" condition [START_REF] Li-Bland | Selective categories and linear canonical relations[END_REF]22] in a selective category, which in turn is motivated by the drive to build a quantising functor from the category of canonical relations between symplectic manifolds to a category of quantum morphisms. We hope to explore these relations in future work.

Thus we begin in Section 2 with the general concepts of a locality set and locality map, emphasising examples on lattice cones and meromorphic functions, while mentioning several other examples in passing. In Section 3, we equip a locality set with a compatible associative multiplication to give a locality semigroup, locality monoid and locality group. Then through the intermediate structure of a locality vector space, we obtain a locality algebra and further a locality Rota-Baxter algebra. In Section 4, we consider the coalgebraic aspect of the construction which begins with the preliminary but subtle notion of locality tensor product. With it, we introduce the concepts of locality coalgebra, the convolution product for maps from a locality coalgebra to a locality algebra. At this point we can give our first main result, Theorem 4.10, addressing the Locality Conservation Principle in Problem 1.3. Finally in Section 5, we bring the locality algebra and locality coalgebra together to form a locality bialgebra and further a locality Hopf algebra under an extra connectedness condition. Then we prove our second main result, Theorem 5.8, addressing the Locality Product Conservation Principle in Problem 1.4. Both results are applied to the example of the exponential generating sum S : QC → M Q , showing that the orthogonality property and the partial multiplicativity on orthogonal pairs of convex lattice cones are indeed preserved by the renormalised version of S. Notations. Unless otherwise specified, all algebras are taken to be unitary commutative over a field K, and linear maps and tensor products are taken over K. A nonunitary algebra means one which does not necessarily have a unit. The same applies to coalgebras.

Locality for sets and maps

2.1. Concepts of locality sets and locality maps. We begin with a set with an independent relation. Definition 2.1.

(i) An locality set is a couple (X, ⊤) where X is a set and ⊤ ⊆ X × X is a symmetric relation on X, referred to as the locality relation (or independence relation) of the locality set. So for

x 1 , x 2 ∈ X, denote x 1 ⊤x 2 if (x 1 , x 2 ) ∈ ⊤.
When the underlying set X needs to be emphasised, we use the notation X × ⊤ X or ⊤ X for ⊤. (ii) For any subset U of a locality set (X, ⊤), let

U ⊤ := {x ∈ X | (U, x) ⊆ X × ⊤ X} be the polar subset of U.
Remark 2.2.

(i) Thus a locality set is simply a set with a binary symmetric relation: we use the term locality set to be consistent with the derived terminology to be introduced later for various algebraic structures built on top of the locality set. (ii) The binary relation ⊤ plays two roles in our study which are related and yet complementary. On the one hand, it serves as a condition under which two elements are related in various ways (independent, orthogonal, etc.), hence the symmetry requirement. As noted in the introduction, the locality relation is intended to encode the notion of independence of events in physics (thus the alternative name independence relation); on the other hand, it assigns the subset of X × X as the domain for partial binary operations in the context of universal algebra [12]. Strictly speaking, the symmetric condition is not required for the latter purpose even though in most of our applications, the algebras are commutative and hence the domain is symmetric. As the two roles are so closely related, we will only deal with symmetric relations unless otherwise needed.

From a locality set one can derive other locality sets as follows.

Lemma 2.3. Let (X, ⊤) be a locality set.

(i) For a subset X ′ of X, denote ⊤ ′ := (X ′ × X ′ ) ∩ ⊤.
Then the pair (X ′ , ⊤ ′ ) is a locality set, called a sub-locality set of (X, ⊤); (ii) For subsets A and B of (X, ⊤), denote A⊤ P B, or simply A⊤B should the context be clear, if A × B ⊆ ⊤. Then ⊤ P equips the power set P(X) of X with a locality set structure; (iii) Combining the above two items, any subset Y of P(X) with the restriction of ⊤ P defines a locality set (Y, (Y × Y ) ∩ ⊤ P ).

As a very simple yet fundamental example which underlies the more sophisticated examples to be discussed below, the orthogonality relation ⊥ Q between vectors or subsets in an Euclidean vector space (V, Q) equips V or the set of subsets of V with the structure of a locality set.

Definition 2.4. A locality map from a locality set (X, ⊤ X ) to a locality set (Y, ⊤ Y ) is a map φ : X → Y such that (φ × φ)(⊤ X ) ⊆ ⊤ Y . More generally, maps φ, ψ : (X, ⊤ X ) → (Y, ⊤ Y ) are called independent and denoted φ⊤ψ if (φ × ψ)(⊤ X ) ⊆ ⊤ Y . To be specific, if φ = ψ, φ is called independent of ψ.
Example 2.5. Any orthogonal map between two Euclidean vector spaces (V i , Q i ), i = 1, 2, is a locality map between the locality spaces (V i , ⊥ Q i ).

Remark 2.6.

(i) The identity map on a locality set (X, ⊤) is trivially a locality map. Also, the composition of two locality maps is still a locality map. Thus locality sets and locality maps form a category. (ii) A map independent of the identity is a locality map. Indeed let (Ω, ⊤ Ω ) be a locality set and φ : Ω → Ω be a map such that φ⊤Id Ω . Then for any (x, y) ∈ ⊤ Ω we have

φ(x)⊤ Ω Id Ω (y) ⇒ y⊤ Ω φ(x) ⇒ φ(y)⊤ Ω Id Ω (φ(x)) ⇒ φ(x)⊤ Ω φ(y).
Note that here we need the symmetric condition.

2.2.

Examples of locality sets and locality maps.

2.2.1.

Convex lattice cones and meromorphic functions. We now give some background for the main examples which serve as both the motivation and prototype of the theoretical structures in this paper. See [15,[START_REF] Guo | A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles[END_REF] for details.

Our first example of locality sets is given by convex polyhedral lattice cones. Consider the filtered rational Euclidean lattice space In (R ∞ , Z ∞ , Q), we write ⊥ Q for the corresponding orthogonality relation.

R ∞ = ≥1 R k , Z ∞ = ≥1 Z k , Q = (Q k (•, •)) k≥1 , where Q k (•, •) : R k ⊗ R k → R, k ≥ 1, is an inner product in R k such that Q k+1 | R k ⊗R k = Q k and Q k (Z k ⊗ Z k ) ⊂ Q, A lattice cone is a pair (C, Λ C ) where C is a polyhedral cone in some R k , that is, C = u 1 , • • • , u m := m i=1 c i u i c i ∈ R ≥0 , 1 ≤ i ≤ m for some u 1 , • • • , u m ∈ Q k ,
Definition 2.7. We call two lattice cones (C, Λ C ) and (D,

Λ D ) orthogonal (with respect to Q), if Q(u, v) = 0 for all u ∈ Λ C , v ∈ Λ D . Then we write (C, Λ C ) ⊥ Q (D, Λ D ).
Multivariate meromorphic functions provide another fundamental motivation for the

forthcoming algebraic setup. Again in (R ∞ , Z ∞ , Q), let M Q ((R k ) * ⊗ C
) be the space of meromorphic germs at 0 with linear poles and rational coefficients [15, 16] and let ( 5)

M Q := k≥1 M Q ((R k ) * ⊗ C).
An element of M Q can be written as a sum of a holomorphic germ and elements the form

(6) h(ℓ 1 , • • • , ℓ m ) L s 1 1 • • • L sn n , s 1 , • • • , s n ∈ Z >0 ,
where h is a holomorphic germ with rational coefficients in linear forms

ℓ 1 , • • • , ℓ m ∈ Q k , and L 1 , • • • , L n are linearly independent linear forms in Q k , ℓ i ⊥ Q L j for all i ∈ {1, • • • , m} for all j ∈ {1, • • • , n}, which is called a polar germ.
Definition 2.8. Two meromorphic germs with rational coefficients f and f ′ are Qorthogonal which we denote by f

⊥ Q f ′ if there exist linear functions L 1 , • • • , L m ∈ Q k and L ′ 1 , • • • , L ′ n ∈ Q k satisfying Q(L i , L ′ j ) = 0 for i = 1, • • • , m, j = 1, • • • , n, and mero- morphic germs g ∈ M Q (R m ⊗ C) and g ′ ∈ M Q (R n ⊗ C), such that f = g(L 1 , • • • , L m ), f ′ = g ′ (L ′ 1 , • • • , L ′ n ). Let (M Q , ⊥ Q
) denote the resulting locality set. We next give examples of locality maps. Let (C, Λ C ) be a strongly convex lattice cone in R k with interior C o . For z in the dual cone

C -:= {z ∈ (R k ) * | x, z < 0, ∀x ∈ C},
we define its exponential generating function to be the sum

S(C, Λ C )(z) := n∈C o ∩Λ C e n,z .
We also define its exponential integral I(C, Λ C ) to be the integral

I(C, Λ C )(z) := C e x,z dΛ x ,
where dΛ x is the volume form induced by generators of Λ C such that the polytope generated by a basis of Λ C has volume 1.

These assignments extend by subdivisions to maps:

(7) S, I : C → M Q .
Proposition 2.9. For lattice cones (C, Λ C ) and

(D, Λ D ), if (C, Λ C ) ⊥ Q (D, Λ D ), then S(C, Λ C )(z) ⊥ Q S(D, Λ D )(z) and I(C, Λ C )(z) ⊥ Q I(D, Λ D )(z)
in the sense of Definition 2.8, that is, the exponential integral and exponential generating function maps I and S are locality maps.

Proof. For any subdivision

{(C i , Λ C i )} of (C, Λ C ), since Λ C i = Λ C , we know (C i , Λ C i ) ⊥ Q (D, Λ D ).
Because any lattice cone can be subdivided into smooth lattice cones, we can reduce the proof to smooth lattice cones. For a smooth lattice cone (C,

Λ C ) = ( u 1 , • • • , u n , n i=1 Zu i ), we have S(C, Λ C )(z) = n i=1 e u i ,z 1 -e u i ,z ; I(C, Λ C )(z) = k i=1 1 u i , z . So S(C, Λ C )(z) and S(D, Λ D )(z) (resp. I(C, Λ C )(z) and I(D, Λ D )(z)) are meromorphic germs in perpendicular linear functions. Therefore S(C, Λ C )(z) ⊥ Q S(D, Λ D )(z) and I(C, Λ C )(z) ⊥ Q I(D, Λ D )(z).

Other examples.

There are many other examples of locality sets. To save space, we only briefly list some of them and refer the reader to the references for further details.

A large number of examples come from disjointness of subsets noted in Lemma 2.3. (i) On the one hand, locality structures can be built on functions or distributions by requiring the disjointness of (adequately chosen) supports of such maps, such as their ordinary supports, their singular supports or wavefront sets (see [START_REF] Brouder | A smooth introduction to the wavefront set[END_REF] and references therein). (ii) On the other hand, locality structures on maps can be built by requiring disjointness of their image sets. This is the case for the decorating maps from the vertices of graphs or trees to a decorating set, which yields locality sets of labelled graphs and trees [11]. (iii) When the set S is equipped with a linear structure, we can replace disjointness by trivial intersections, transversality or linear independence. If S further has an inner product, then the relation can be chosen to be the one of orthogonality. Further examples of locality sets include independence of events in probability and coprimeness of natural numbers as discussed in the introduction.

Building up locality from semigroups to Rota-Baxter algebras

In this section, we equip a locality set with various algebraic structures, from that of a semigroup to that of a Rota-Baxter algebra.

3.1. Locality semigroups. For a locality set (X, T ) and an integer k ≥ 2, denote

(8) X ⊤ k := X × ⊤ • • • × ⊤ X k factors := (x 1 , • • • , x k ) ∈ X k | (x i , x j ) ∈ ⊤, 1 ≤ i = j ≤ k . Definition 3.1.
(i) An locality semigroup2 is a locality set (G, ⊤) together with a product law defined on ⊤:

m G : G × ⊤ G -→ G
for which the product is compatible with the locality relation on G, namely [START_REF] Dang | Complex powers of analytic functions and meromorphic renormalization in QFT[END_REF] for all

U ⊆ G, m G ((U ⊤ × U ⊤ ) ∩ ⊤) ⊂ U ⊤
and such that the following locality associativity property holds:

(10) (x • y) • z = x • (y • z) for all (x, y, z) ∈ G × ⊤ G × ⊤ G.
Note that, because of the condition ( 9), both sides of Eq. ( 10) are well-defined for any triple in the given subset.

(ii) An locality semigroup is commutative if m G (x, y) = m G (y, x) for (x, y) ∈ ⊤,
noting that both sides of the equations are defined since ⊤ is symmetric. (iii) An locality monoid is a locality semigroup (G, ⊤, m G ) together with a unit element 1 G ∈ G given by the defining property

{1 G } ⊤ = G and m G (x, 1 G ) = m G (1 G , x) = x for all x ∈ G.
We denote the locality monoid by (G,

⊤, m G , 1 G ). (iv) An locality group is a locality monoid (G, ⊤, m G , 1 G ) equipped with a morphism ι : G → G of locality sets, called the inverse map, such that (ι(g), g) ∈ ⊤ and m G (ι(g), g) = m G (g, ι(g)) = 1 G for any g ∈ G. (v) A sub-locality semigroup of a locality semigroup (G, ⊤, m G ) is a locality semi- group (G ′ , ⊤ ′ , m G ′ ) with G ′ ⊆ G, ⊤ ′ = (G ′ × G ′ ) ∩ ⊤ and m G ′ = m G | ⊤ ′ , that is, for x, y ∈ G ′ and (x, y) ∈ ⊤, m G (x, y) is in G ′ .
A sub-locality monoid of a locality monoid is a sub-locality semigroup of the corresponding locality semigroup which share the same unit. A sub-locality group of a locality group is a sub-locality monoid of the corresponding locality monoid which is also a locality group.

For notational convenience, we usually abbreviate m G (x, y) by x • y or simply xy.

Remark 3.2. One easily checks that on a locality monoid (G,

⊤, m G , 1 G ) if (x 1 , x 2 , y 1 , y 2 ) is in G ⊤ 4 , then (x 1 x 2 ,
Q : M Q × ⊥ Q M Q → M Q , is a locality monoid. Definition 3.5. Let (X, ⊤ X , • X ) and (Y, ⊤ Y , • Y ) (resp. (X, ⊤ X , • X , 1 X ) and (Y, ⊤ Y , • Y , 1 Y ))
be locality semigroups (resp. monoids). A map φ : X -→ Y is called a locality semigroup (resp. locality monoid) homomorphism, if it (i) is a locality map;

(ii) is locality multiplicative: for (a, b) ∈ ⊤ X we have φ(a

• X b) = φ(a) • Y φ(b), (iii) (resp. preserves the unit φ(1 X ) = 1 Y .)
Example 3.6. Classical examples of locality monoid homomorphisms are given by multiplicative functions in number theory. Here a function f :

Z ≥1 → Z ≥1 is multiplicative means f (1) = 1 and f (mn) = f (m)f (n) if m
and n are coprime. This means precisely that f is a locality monoid homomorphism from the locality monoid (Z ≥1 , ⊤ cop ) where ⊤ cop is the coprime relation, to the locality monoid (Z ≥1 , ⊤ full ), where ⊤ full is the full relation

Z ≥1 × Z ≥1 .
Now let us take a closer look at the set C of lattice cones. For convex cones

C := u 1 , • • • , u m and D := v 1 , • • • , v n spanned by u 1 , • • • , u m and v 1 , • • • , v n respectively, their Minkowski product (usually called Minkowski sum) is the convex cone C • D := u 1 , • • • , u m , v 1 , • • • , v n .
This product can be extended to a product in C:

(11) (C, Λ C ) • (D, Λ D ) := (C • D, Λ C + Λ D ),
where Λ C +Λ D is the abelian group generated by Λ C and Λ D in Q ∞ . This product endows a monoid structure on C with unit ({0}, {0}), which also restricts to a locality monoid structure on (C, ⊥ Q ).

Even though C and M Q have their natural multiplications defined on the full spaces, the importance of the locality structures on C and M Q becomes evident when studying the multiplicative property of the maps I and S from C to M Q introduced in Section 2.2.

Because of the idempotency (C, Λ

C ) • (C, Λ C ) = (C, Λ C ) for (C, Λ C ) ∈ C, the multi- plicativity I((C, Λ C ) • (C, Λ C ))(z) = I(C, Λ C )(z)I(C, Λ C )(z) or S((C, Λ C ) • (C, Λ C ))(z) = S(C, Λ C )(z)S(C, Λ C )(z)
does not hold in general since that would force the integral or the sum to be 0 or 1, which can not be the case for example by taking (C, Λ C ) = ( e 1 , Ze 1 ). But the multiplicativity can be recovered in the context of locality monoids, as follows.

Proposition 3.7. If (C, Λ C ) ⊥ Q (D, Λ D ), then S((C, Λ C ) • (D, Λ D )) = S(C, Λ C )S(D, Λ D ), I((C, Λ C ) • (D, Λ D )) = I(C, Λ C )I(D, Λ D ).
Thus the locality maps I and S are locality semigroup homomorphisms from

(C, ⊥ Q ) to (M Q , ⊥ Q ). Proof. For (C, Λ C ) ⊥ Q (D, Λ D ), if both are smooth, let C := u 1 , • • • , u m with Λ C = ⊕Zu i , and D := v 1 , • • • , v n with Λ D = ⊕Zv j . Since Q(u i , v j ) = 0, we have Λ C + Λ D = ⊕ i Zu i ⊕ ⊕ j Zv j . Thus (C, Λ C ) • (D, Λ D ) is smooth. By a direct calculation, we obtain S ((C, Λ C ) • (D, Λ D )) = S(C, Λ C ) S(D, Λ D ), I ((C, Λ C ) • (D, Λ D )) = I(C, Λ C I(D, Λ D ).
The general case follows by a subdivision of lattice cones into smooth lattice cones.

Moreover, the locality monoid (C, ⊥ Q ) has a natural grading which does not extend to the monoid C. For n ≥ 0, let C n denote the subset of C consisting of lattice cones of dimension n. Then for orthogonal lattice cones (C, Λ C ) and (D, Λ D ), we have dim

((C, Λ C )• (D, Λ D )) = dim(C, Λ C ) + dim(D, Λ D ). Hence, (12) C = ⊔ n≥0 C n , m C ((C m × C n )∩ ⊥ Q ) ⊆ C m+n for all m, n ≥ 0,
which we take to be the defining conditions for C to be a graded locality monoid.

3.2. Locality vector spaces. We now consider locality relations on vector spaces.

Definition 3.8. An locality vector space is a vector space V equipped with a locality relation ⊤ which is compatible with the linear structure on V in the sense that, for any subset X of V , X ⊤ is a linear subspace of V .

Remark 3.9. For a locality vector space (V, ⊤), since V ⊤ is a linear subspace of V , we have {0} × V ⊂ ⊤, or equivalently 0 ∈ V ⊤ . Note that there is no locality restrictions for the vector space structure (addition and scalar product) on V , that is, the addition and scalar product are everywhere defined.

Example 3.10. The vector space M Q equipped with the relation ⊥ Q in Definition 2.8 is a locality vector space M Q , ⊥ Q .

Remark 3.11. Clearly, constant germs are orthogonal of any germs, namely R⊆M ⊥ Q Q . In fact, the converse is also true. Thus M ⊥ Q Q = R. From a locality set (X, ⊤) we can build a locality vector space (KX, ⊤) from the vector space generated by X whose defining relation (denoted by the same symbol ⊤) is the linear extension of that on X. More precisely for u, v ∈ KX, (u, v) ∈ ⊤ if the basis elements from X appearing in u are related via ⊤ to the basis elements appearing in v.

Thus KX × ⊤ KX KX = U,V ⊆X,(U,V )⊆⊤ KU × KV.
Example 3.12. The locality set C of lattice cones with the orthogonal relation in Definition 2.7 gives rise to the corresponding locality vector space QC. Likewise, the locality set of labelled rooted trees described in Section 2.2.2 gives rise to the corresponding locality vector space generated by the set.

Definition 3.13. Let (U, ⊤ U ) and (V, ⊤ V ) be locality vector spaces, a linear map φ : (U, ⊤ U ) → (V, ⊤ V ) is called a locality linear map if it is a locality map.

Example 3.14. The locality maps given by the exponential integral I : C → M Q and the exponential generating sum S : C → M Q in Proposition 2.9 extend to locality linear maps from QC to M Q .

Here are further useful properties of locality linear maps.

Proposition 3.15. Let (U, ⊤ U ), (V, ⊤ V ) be locality vector spaces and φ, ψ : (U, ⊤ U ) -→ (V, ⊤ V ) be independent locality linear maps. Any two linear combinations of φ and ψ are also independent. In particular, any linear combination λφ+µψ with λ, µ ∈ K is a locality linear map.

Proof. Let u 1 , u 2 be in ⊤ U . Since φ and ψ are independent locality linear maps, we have {φ(u 1 ), ψ(u 1 )}⊤ V {φ(u 2 ), ψ(u 2 )} and hence (λφ(u 1 ) + µψ(u 1 )) ⊤ V (λφ(u 2 ) + µψ(u 2 )) .

Locality algebras.

We begin with a preliminary notion. Let V and W be vector spaces and let

⊤ := V × ⊤ W ⊆ V × W . A map f : V × ⊤ W → U to a vector space U is called locality bilinear if f (v 1 + v 2 , w 1 ) = f (v 1 , w 1 ) + f (v 2 , w 1 ), f (v 1 , w 1 + w 2 ) = f (v 1 , w 1 ) + f (v 1 , w 2 ), f (kv 1 , w 1 ) = kf (v 1 , w 1 ), f (v 1 , kw 1 ) = kf (v 1 , w 1 ) for all v 1 , v 2 ∈ V , w 1 , w 2 ∈ W and k ∈ K such
that all the pairs arising in the above expressions are in V × ⊤ W . Definition 3.16.

(i) A nonunitary locality algebra over K is a locality vector space (A, ⊤) over K together with a locality bilinear map

m A : A × ⊤ A → A such that (A, ⊤, m A )
f (u • A v) = f (u) • B f (v) for all (u, v) ∈ ⊤ A .
(vi) A locality algebra A with a linear grading

A = ⊕ n≥0 A n is called a locality graded algebra if m A ((A m × A n ) ∩ ⊤ A ) ⊆ A m+n for all m, n ∈ Z.
It is easy to check that a locality linear map f : (A, ⊤ A , • A ) → (B, ⊤ B , • B ) between two locality algebras is a locality algebra homomorphism if and only if ker f is a locality ideal of A, by the same argument as the one for the corresponding result on an algebra homomorphism.

Remark 3.17.

(i) For a locality algebra (A, ⊤) we have {0,

1 A } ⊂ A ⊤ since 0 ∈ A ⊤ by Remark 3.9. (ii) If A × ⊤ A is A × A in
a locality monoid and locality algebra, we recover the usual notions of monoid and algebra.

The locality space QC, with the multiplication obtained from the linear extension of the locality monoid structure on C by the Minkowski product in Eq. ( 11), is a locality commutative algebra. By Eq. (12), we in fact have Lemma 3.18. With the grading induced from C = ⊔ n≥0 C n in Eq. (12), QC = ⊕ n≥0 QC n is a graded locality algebra.

Another important locality algebra for our purpose is the space M Q , ⊥ Q with ⊥ Q as in Definition 2.8 and the pointwise multiplication. Further by Proposition 3.7, the linear maps [START_REF] Guo | Renormalization of multiple zeta values[END_REF] S, I : QC → M Q linearly extended from those in Eq. ( 7), are locality algebra homomorphisms. We can say more about the locality algebra M Q . By [START_REF] Guo | A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles[END_REF]Corollary 4.7], there is a direct sum decomposition (15)

M Q = M Q,+ ⊕ M Q Q,-
, where M Q,+ is the subspace of holomorphic functions and M Q Q,-is the subspace spanned by polar germs defined by Eq. ( 6). Then we have the following result from [START_REF] Guo | A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles[END_REF]Corollary 4.18] reformulated in the terminology of locality structures.

Proposition 3.19. In the decomposition in Eq. (15), the space M Q,+ is a subalgebra and a locality subalgebra of M Q . The space M Q Q,-is not a subalgebra but a locality subalgebra, in fact a locality ideal of M Q . Consequently, the projection

π Q + : M Q → M Q,+ is a locality algebra homomorphism.
In contrast to the multivariate case, the space

M Q Q,-(R * ⊗ C) = ǫ -1 C[ǫ -1
] is a subalgebra in the space M Q (R * ⊗ C) of meromorphic functions in one variable. This is a major difference between our multivariate setup and the usual single variate framework used for renormalisation purposes. We circumvent the difficulty in relaxing ordinary multiplicativity to a multiplicativity allowed only on independent elements. In fact,

M Q Q,-(R * ⊗C) is a locality ideal of M Q (R * ⊗C) under the restriction of independence relation since the local- ity relation ⊥ Q restricted to M Q (R * ⊗C) is simply (C×M Q (R * ⊗C))∪(M Q (R * ⊗C)×C).
Thus, the locality algebra homomorphism π Q + restricts to a mere linear map on M Q (R * ⊗C) with no additional multiplicativity property.

3.4. Locality Rota-Baxter algebras and projection maps. The reader is referred to [START_REF] Guo | An introduction to Rota-Baxter algebra[END_REF] for background on Rota-Baxter algebras.

Definition 3.20. A linear operator P : A → A on a locality algebra (A, ⊤) over a field K is called a locality Rota-Baxter operator of weight λ ∈ K, or simply a Rota-Baxter operator, if it is a locality map, independent of Id A , and satisfies the following locality Rota-Baxter relation: We call the triple (A, ⊤, P ) a locality Rota-Baxter algebra.

Remark 3.21.

(i) The right hand side of Eq. ( 16) is well defined due to the condition that P is independent of the identity. (ii) As in the classical setup [13, Proposition 1.1.12], if P is a locality Rota-Baxter operator of weight λ, then -λ -P is also a locality Rota-Baxter of weight λ.

An important class of locality Rota-Baxter algebras arises from idempotent operators.

Proposition 3.22. Let (A, ⊤, m A ) be a locality algebra. Let P : A -→ A be a locality linear idempotent operator in which case there is a linear decomposition A = A 1 ⊕ A 2 with A 1 = ker (Id -P ) and A 2 = ker (P ) so that P is the projection onto A 1 along A 2 . The following statements are equivalent: (i) P or Id -P is a locality Rota-Baxter operator of weight -1;

(ii) A 1 and A 2 are locality subalgebras of A, and P and Id -P are independent locality maps.

If one of the conditions holds, then P is a locality multiplicative map if and only if A 2 is a locality ideal of A.

Proof. We write π 1 = P and π 2 = Id -P . ((i) =⇒ (ii)) It follows from the locality Rota-Baxter identity ( 16) that A 1 = P (A) is a sub-locality algebra of A. Since Id -P is again an idempotent locality Rota-Baxter operator, A 2 = (Id -P )(A) is also a sub-locality algebra of A. Finally, P and Id -P are independent locality maps as a consequence of Definition 3.20. ((ii) =⇒ (i)) Since π 1 and π 2 = Idπ 1 are locality Rota-Baxter operators of weight -1 at the same time in view of Remark 3.21, we only need to verify that π 1 is a locality Rota-Baxter operator of weight -1:

(17) π 1 (a) π 1 (b) + π 1 (a b) = π 1 (π 1 (a) b) + π 1 (a π 1 (b)) for all (a, b) ∈ ⊤.
Write a = a 1 + a 2 and b = b 1 + b 2 . Since the projections π i , i = 1, 2, are independent locality maps, it follows that {a 1 , a 2 }⊤{b 1 , b 2 }. Thus every term in

ab = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2 is well defined, with a 1 b 1 ∈ A 1 and a 2 b 2 ∈ A 2 .
Then the left hand side of Eq. ( 17) becomes

a 1 b 1 + π 1 (a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2 ) = 2a 1 b 1 + π 1 (a 1 b 2 ) + π 1 (a 2 b 1 ).
The right hand side of Eq. ( 17) becomes

π 1 (a 1 b) + π 1 (ab 1 ) = π 1 (a 1 b 1 + a 1 b 2 ) + π 1 (a 1 b 1 + a 2 b 1 ) = π 1 (a 1 b 2 ) + π 1 (a 2 b 1 ) + 2a 1 b 1 ,
as needed. Then the last statement follows from the remark after Definition 3.16.

As an immediate consequence of Proposition 3.19 and 3.22, we obtain Corollary 3.23. The projection

π Q + onto M Q,+ along M Q Q-is locality multiplicative and (M Q , π Q -)
is a locality Rota-Baxter algebra.

Locality for coalgebras and the Locality Conservation Principle

We introduce the concept of a locality coalgebra which involves a suitable locality tensor product. Between a locality coalgebra and a locality algebra, we consider locality convolution and locality convolution inverse when the locality coalgebra is connected. We then prove our Locality Conservation Principle, showing that a locality (independent) relation is preserved by the renormalisation procedure à la Connes and Kreimer i.e., carried out by means of algebraic Birkhoff factorisation in the coalgebra-algebraic context [15]. 4.1. Locality tensor product. We first give a locality version of the tensor product by considering a relative generalisation of the locality relation. As noted before, vector spaces and tensor products are taken over the base field K unless otherwise specified.

Let V and W be vector spaces and let ⊤ ⊆ V × W . For X ⊆ V and Y ⊆ W , denote

X ⊤ := {w ∈ W | X⊤w}, ⊤ Y := {v ∈ V | v⊤Y }.
A relative locality vector space is a triple (V, W, ⊤) where V and W are vector spaces and ⊤ := V × ⊤ W is a subset of V × W such that for any sets X ⊆ V and Y ⊆ W , the sets X ⊤ and ⊤ Y are linear subspaces of V and W respectively.

Example 4.1. Given vector spaces V and W , any subspaces V 1 ⊂ V and W 1 ⊂ W give rise to a relative locality vector space (V, W,

⊤) with ⊤ = (V 1 ×W 1 )∪({0}×W )∪(V ×{0}).
Given two vector spaces (resp. a relative locality vector space (V, W, ⊤)), define I bilin (resp. I ⊤,bilin ) to be the subspace of K(V × W ) (resp. K(V × ⊤ W )) spanned by the bilinear relations

(18) (v 1 + v 2 , w 1 ) -(v 1 , w 1 ) -(v 2 , w 1 ), (v 1 , w 1 + w 2 ) -(v 1 , w 1 ) -(v 1 , w 2 ), (kv 1 , w 1 ) -(v 1 , kw 1 ), (kv 1 , w 1 ) -k(v 1 , w 1 ) for all v 1 , v 2 ∈ V, w 1 , w 2 ∈ W, k ∈ K (resp.
such that the pairs in Eq. ( 18) are in ⊤).

Counterexample 4.2. Take {e 1 , e 2 , e 3 , e 4 } to be the canonical orthonormal basis of V := R 4 and

V × ⊤ V = ((V × {0}) ∪ ({0} × V )) ∪ (K (e 1 + e 2 ) × K e 3 ) ∪ (K (e 1 + 2e 2 ) × K e 4 ) ∪ (K e 1 × K (e 3 + e 4 )) ∪ (K e 2 × K (e 3 + 2e 4 )) .
Then (-e 1e 2 , e 3 ) + (-e 1 -2e 2 , e 4 ) + (e 1 , e 3 + e 4 ) + (e 2 , e 3 + 2e 4 ) is an element of K(V × ⊤ W ) ∩ I bilin as can easily be seen using the defining relations for I bilin , but it is not in I ⊤,bilin .

So in general

K(V × ⊤ W ) ∩ I bilin I ⊤,bilin , and thus h :

K(V × ⊤ W )/I ⊤,bilin → V ⊗ W is not injective. Therefore K(V × ⊤ W )/I ⊤,
bilin is not an appropriate candidate for a locality tensor product, since the image of a locality coproduct ∆ should lie in V ⊗ V . Instead, we set

(19) V ⊗ ⊤ W := im h ⊂ V ⊗ W,
which is also the image of the composition map

K(V × ⊤ W ) → K(V × W ) → V ⊗ W.
4.2. Locality coalgebras. We recall that a coalgebra (C, ∆) over a field K is counital if there is a map ε :

C → K such that (ε⊗Id C )∆ = (Id C ⊗ε)∆ = Id C . It is (Z ≥0 -)graded if C = n∈Z ≥0 C n and ∆(C n ) ⊆ p+q=n C p ⊗ C q , n≥1 C n ⊆ ker ε. Thus C = C 0 + ker ε. Moreover a graded coalgebra is called connected if C = C 0 ⊕ ker ε.
Consequently, ε restricts to a linear bijection ε :

C 0 ∼ = K and ker ε = ⊕ n≥1 C n .
For the sake of simplicity, we shall drop the explicit mention of the grading and simply call such a coalgebra a connected coalgebra.

The following definition is dual to that of a locality algebra. As in Eq. ( 19), let

C ⊗ ⊤ n = C ⊗ ⊤ • • • ⊗ ⊤ C n factors denote the image of KC ⊤ n in C ⊗n .
Definition 4.3. Let (C, ⊤) be a locality vector space and let ∆ : C → C ⊗ C be a linear map. (C, ⊤, ∆) is a locality noncounital coalgebra if it satisfies the following two conditions (i) for any U ⊂ C (compare with Eq. ( 9))

(20) ∆(U ⊤ ) ⊂ U ⊤ ⊗ ⊤ U ⊤ .
In particular, ∆(C) ⊆ C ⊗ ⊤ C; (ii) the following coassociativity holds:

(Id C ⊗ ∆) ∆ = (∆ ⊗ Id C ) ∆.
• If in addition, there is a counit, namely a linear map ε :

C → K such that (Id C ⊗ ε) ∆ = (ε ⊗ Id C ) ∆ = Id C , then (C, ⊤, ∆, ε) is called a locality coalgebra. • A connected locality coalgebra is a locality coalgebra (C, ⊤, ∆) with a grading C = ⊕ n≥0 C n such that, for any U ⊆ C, (21)
∆(C n ∩ U ⊤ ) ⊆ p+q=n (C p ∩ U ⊤ ) ⊗ ⊤ (C q ∩ U ⊤ ), n≥1 C n = ker ε.
We denote by J the unique element of C 0 with ε(J) = 1 K , giving C 0 = K J.

Remark 4.4. We note that whereas the conditions for a locality algebra are weaker than those for an algebra, the conditions for a locality coalgebra are stronger than those for a coalgebra. In particular, a locality coalgebra is a coalgebra and a connected locality coalgebra is a connected coalgebra.

A natural example of locality coalgebra is given by the coalgebra QC of lattice cones in [15]. Let (C, Λ C ) be a lattice cone, (F, Λ F ) a face of (C, Λ C ), which we denote by (F, Λ F ) < (C, Λ C ). The transverse cone t((C, Λ C ), (F, Λ F )) is the orthogonal projection of (C, Λ C ) to the orthogonal subspace of the subspace spanned by F . Then the coproduct ∆(C, Λ C ) of (C, Λ C ) is defined by

(22) ∆((C, Λ C )) = F <C t((C, Λ C ), (F, Λ F )) ⊗ (F, Λ F ). Since t((C, Λ C ), (F, Λ F )) ⊥ Q (F, Λ F ) by definition, the quadruple (QC, ⊥ Q , ∆, ε
) is a locality coalgebra with the locality counit given by the linear extension of the map

ε : C → Q, (C, Λ C ) → 1, (C, Λ C ) = ({0}, {0}), 0, (C, Λ C ) = ({0}, {0}).
Further the connectedness conditions in Eq. ( 21) are satisfied. Thus we have proved Lemma 4.5. (QC, ⊥ Q , ∆, ε) with the grading QC = ⊕ n≥0 QC n from Eq. ( 12) is a connected locality coalgebra.

Let us list a few useful general properties of locality coalgebras.

Lemma 4.6. Let (C, ⊤, ∆) be a locality coalgebra. (i) For any n ≥ 2 and 0

≤ i ≤ n, (23)
Id ⊗i C ⊗ ∆ ⊗ Id ⊗(n-i-1) C : C ⊗ ⊤ n → C ⊗ ⊤ (n+1) (ii) We have (∆ ⊗ ∆)(C ⊗ ⊤ C) ⊆ C ⊗ ⊤ 4 ; (iii) We have (∆ × ∆)(C × ⊤ C) ⊆ (C ⊗ ⊤ C) × ⊤ (C ⊗ ⊤ C);
(iv) For any locality linear map φ independent of Id, n ≥ 2 and 0 ≤ i ≤ n, we have

Id ⊗i ⊗ φ ⊗ Id ⊗(n-i-1) : C ⊗ ⊤ n → C ⊗ ⊤ n .
Proof. (i) Let n ≥ 2 and 1 ≤ i ≤ n be given. By the definition of C ⊗ ⊤ n , any of its elements is a sum of pure tensors

c 1 ⊗ • • • ⊗ c n with (c 1 , • • • , c n ) ∈ C ⊤ n . Let U = {c j | j = i + 1}. Then c i+1 ∈ U ⊤ , so there exist (d 1 , e 1 ), • • • , (d k , e k ) ∈ U ⊤ × ⊤ U ⊤ , such that ∆(c i+1 ) = ℓ d ℓ ⊗ e ℓ . Now (Id ⊗i C ⊗ ∆ ⊗ Id ⊗(n-i-1) C )(c 1 ⊗ • • • ⊗ c n ) = ℓ c 1 ⊗ • • • c i ⊗ d ℓ ⊗ e ℓ ⊗ c i+2 ⊗ • • • ⊗ c n ,
and

c 1 ⊗ • • • c i ⊗ d ℓ ⊗ e ℓ ⊗ c i+2 ⊗ • • • ⊗ c n ∈ C ⊗ ⊤ (n+1) . (ii) Since (∆ ⊗ ∆) = (∆ ⊗ Id)(Id ⊗ ∆)
, from Eq. ( 23) we obtain

(∆ ⊗ ∆)(C ⊗ ⊤ C) = (∆ ⊗ Id)(Id ⊗ ∆)(C ⊗ ⊤ C) ⊆ (∆ ⊗ Id)(C ⊗ ⊤ C ⊗ ⊤ C) ⊆ C ⊗ ⊤ C ⊗ ⊤ C ⊗ ⊤ C. (iii) Let (c 1 , c 2 ) ∈ C × ⊤ C. Then c 2 ∈ {c 1 } ⊤ . So by Eq. (20), ∆(c 2 ) = (c 2 ) c 2,( 1
) ⊗ c 2,(2) ) is in (C ⊗ ⊤ C) × ⊤ (C ⊗ ⊤ C). (iv) Again any element of C ⊗ ⊤ n is a sum of pure tensors c 1 ⊗ • • • ⊗ c n with (c 1 , • • • , c n ) ∈ C ⊤ n . Thus (c 1 , • • • , c i-1 , φ(c i ), c i+1 , • • • , c n ) is in C ⊤ n . This is what we want since (Id ⊗i ⊗ φ ⊗ Id ⊗(n-i-1) )(c 1 ⊗ • • • ⊗ c n ) = c 1 ⊗ • • • ⊗ c i-1 ⊗ φ(c i ) ⊗ c i+1 ⊗ • • • ⊗ c n .
Lemma 4.7. Let (C = ⊕ n≥0 C n , ⊤, ∆) be a connected locality coalgebra. Define the reduced coproduct ∆(c) := ∆(c) -J ⊗ cc ⊗ J. Recursively define

(24) ∆(1) = ∆, ∆(k) := (Id ⊗ ∆(k-1) ) ∆, k ≥ 2. (i) For c ∈ ⊕ n≥1 C n , ∆(c) = (c) c ′ ⊗c ′′ with deg(c ′ ), deg(c ′′ ) > 0 and (c ′ , c ′′ ) ∈ C× ⊤ C; (ii) If in addition c ∈ U ⊤ for some U ⊂ C, then the above pairs (c ′ , c ′′ ) are in U ⊤ × ⊤ U ⊤ ; (iii) ∆(k) (x) is in C ⊗ ⊤ (k+1) for all x ∈ C, k ∈ N; (iv) ∆(k) (C n ) = {0} for all k ≥ n.
Proof. We only need to prove (ii) since then (i) is the special case when U = {0}.

Let c ∈ C n ∩ U ⊤ . By Eq. ( 21), we can write

∆(c) = y ⊗ J + J ⊗ z + (c) c ′ ⊗ c ′′ with y, z ∈ C n , c ′ , c ′′ ∈ U ⊤ and each c ′ ⊗ c ′′ ∈ C p ⊗ ⊤ C q , p + q = n, p, q ≥ 1.
Then by the same argument for a connected coalgebra [13, Theorem 2.3.3], we obtain y = z = x. This proves (ii).

Then (iii) follows from an easy induction on k by the locality property of ∆; while the proof of (iv) is similar to the case without a locality structure [18, Proposition II.2.1]. 4.3. Locality of the convolution product. We show that the locality (independence) of linear maps are preserved under the convolution product.

Lemma 4.8. Let (C, ⊤ C ) be a locality coalgebra with counit ε C : C → K. Let (A, ⊤ A ) be a locality algebra with unit u A : K → A. The map e := u A ε C : C → A is independent to any linear map φ : C -→ A. In particular, the map e is a locality linear map.

Proof. This is because im e = K • 1 A ⊂ A ⊤ A as we can see from Remark 3.17. Now we give a general result. Proposition 4.9. Let (C, ⊤ C , ∆) be a locality coalgebra and let (A, ⊤ A , m A ) be a locality algebra. Let L := L-Hom(C, A) be the space of locality linear maps. Define

⊤ L := {(φ, ψ) ∈ L × L | (φ × ψ)(C × ⊤ C) ⊆ A × ⊤ A} .
(i) For (φ, ψ) ∈ ⊤ L , define the convolution product of φ and ψ by

(25) φ ⋆ ψ : C ∆ C -→ C ⊗ ⊤ C φ⊗ψ -→ A ⊗ ⊤ A m A -→ A.
Then φ ⋆ ψ is a locality linear map and the triple (L, ⊤ L , ⋆) is a locality algebra. (ii) If moreover C is connected then

GL := {φ ∈ L | φ(J) = 1 A }
is a locality group for the convolution product. (iii) Under this same assumption, we have

(φ, ψ) ∈ ⊤ L ∩ (GL × GL) =⇒ (φ ⋆k , ψ ⋆l ) ∈ ⊤ L ∩ (GL × GL)
for all k, l ∈ Z.

Proof.

(i) For (φ, ψ) ∈ ⊤ L , since (φ × ψ)(C × ⊤ C) ⊆ A × ⊤ A, we have (φ ⊗ ψ)(C ⊗ ⊤ C) ⊆ A ⊗ ⊤ A.
Hence the composition in Eq. ( 25) is well-defined, giving a well-defined convolution product.

We next verify that φ ⋆ ψ is a locality linear map. For c 1 ⊤ C c 2 , by Lemma 4.6.(iii), there are finitely many (d i , e i ), (f j , g j ) ∈ C × ⊤ C with (d i , e i , f j , g j ) ∈ C ⊤ 4 , such that

∆(c 1 ) = i d i ⊗ e i and ∆(c 2 ) = j f j ⊗ g j . Then φ ⋆ ψ(c 1 ) = i φ(d i )ψ(e i ) and φ ⋆ ψ(c 2 ) = j φ(f j )ψ(g j ).
From (φ, ψ) ∈ ⊤ L , we obtain (φ

(d i ), ψ(e i ), φ(f j ), ψ(g j )) ∈ A ⊤ 4 . So i φ(d i )ψ(e i ), j φ(f j )ψ(g j ) is in A × ⊤ A and thus (φ ⋆ ψ(c 1 ))⊤ A (φ ⋆ ψ(c 2 )).
Thus we are left to verify the axioms for a locality semigroup: the closeness of U ⊤ L under the convolution product for every U ⊆ L and the associativity.

i : C → K + A i with φ i (J) = 1 A and φ i (ker ε) ⊆ A i , i = 1, 2, such that (28) φ = φ ⋆(-1) 1 ⋆ φ 2 .
The map (φ 1 ) ⋆(-1) is a locality linear map and

φ 1 ⊤ L {φ, φ 2 }, φ ⋆(-1) 1 ⊤ L {φ 1 , φ 2 }. (i) If in addition to the Basic Assumption, A 1 is a sub-locality algebra of A, then φ ⋆(-1) 1 : C → K + A 1 .
(ii) If in addition to the Basic Assumption and Item (i), A 2 is a locality ideal of A, then φ ⋆(-1) 1

= π 1 φ and φ 2 is recursively given by

(29) φ 2 (J) = 1 A , φ 2 (c) = (π 2 φ)(c) - (c) (π 1 φ)(c ′ )φ 2 (c ′′ ) for all c ∈ ker ε, with ∆(c) = (c) c ′ ⊗ c ′′ the reduced coproduct defined in Lemma 4.7. If ψ : (C, ⊤ C ) -→ (A, ⊤ A
) is also a locality linear map independent of φ with ψ(J) = 1 A , then φ i and ψ j are independent for i, j = 1, 2.

Remark 4.11. Theorem 4.10 provides an answer to Problem 1.3: the locality of the renormalised map follows from that of the initial map under the assumption of the theorem. See also Remark 4.12 and its subsequent example.

Proof. The proof of the uniqueness of the maps φ i is the same as the proof [15, Theorem 4.4] for the case of a trivial locality relation i.e., when ⊤ = C × C.

Let n ≥ 1 and c ∈ C n . Since C is a connected locality coalgebra, we can write

∆(c) = J ⊗ c + c ⊗ J + (c) c ′ ⊗ c ′′ with deg(c ′ ), deg(c ′′ ) > 0 and c ′ ⊤ C c ′′ .
We first prove by induction on the degree n of c that the map given by (30)

φ 1 (c) = 1 A , c = J, -π 1 φ(c) + (c) φ 1 (c ′ )φ(c ′′ ) , c ∈ C n , n > 0, with c ′ ⊤ C c ′′ of degree ≤ n. By the definition of φ 1 , φ 1 (c) = -π 1 (φ(c) + (c) φ 1 (c ′ ) φ(c ′′ ))
and φ 1 (c ′ )⊤ A φ(c ′′ ). By the assumption on π i , we have φ

1 (c ′ )⊤ A {π 1 φ(c ′′ ), (π 2 φ)(c ′′ )}. Then φ 1 (c) = -π 1 φ(c) - (c) φ 1 (c ′ ) (π 1 φ)(c ′′ ) + (π 2 φ)(c ′′ ) = -(π 1 φ)(c) - (c) φ 1 (c ′ )(π 1 φ)(c ′′ ). Consequently, φ 1 ⋆ (π 1 φ) (c) = φ 1 (c) + π 1 φ(c) + (c) φ 1 (c ′ ) π 1 φ(c ′′ ) = 0. We conclude that φ 1 ⋆ (π 1 φ) = e, leading to φ ⋆(-1) 1 = π 1 φ since φ ⋆(-1) 1
has been shown to exist. The locality algebraic Birkhoff factorization φ = φ ⋆-1 1 ⋆ φ 2 then yields for c∈ ker(ε):

π 1 (φ(c)) + φ 2 (c) + (c) π 1 (φ(c ′ ))φ 2 (c ′′ ) = φ(c)
Using π 2 = Idπ 1 gives the recursive expression (29) for φ 2 in terms of π 1 and π 2 .

Remark 4.12. That the locality of φ implies that of φ 2 and φ 1 can be summarised under the motto "renormalisation preserves locality" in analogy with quantum field theory.

We illustrate this motto by applying Theorem 4.10 to the locality linear map S : QC → M Q that we have been taken as the main example throughout this paper.

(C, Λ C ) ⊥ Q S ⋆(-1) 1 (D, Λ D ), S 2 (C, Λ C ) ⊥ Q S 2 (D, Λ D ).

Locality for Hopf algebras and the Locality Product Conservation Principle

In this section, we combine a locality algebra and a locality coalgebra to give a locality bialgebra, and then a locality Hopf algebra under an additional connectedness condition. Together with the locality Rota-Baxter algebra structure, we obtain the locality version of the algebraic Birkhoff factorisation originally given by Connes and Kreimer. This provides an answer to the question addressed in the Locality Product Conservation Principle in Problem 1.4. An application is given to the renormalisation of the exponential generating function of lattice cones. (i) An locality bialgebra is a sextuple (B, ⊤, m, u, ∆, ε) consisting of a locality algebra (B, m, u, ⊤) and a locality coalgebra (B, ∆, ⊤, ε) that are locality compatible in the sense that ∆ and ε are locality algebra homomorphisms. (ii) A locality bialgebra B is called connected if there is a Z ≥0 -grading B = ⊕ n≥0 B n with respect to which B is both a locality graded algebra in the sense of Definition 3.16 and a connected locality coalgebra in the sense of Definition 4.3. Then J = 1 B .

Let us go back to the space QC of lattice cones with the Minkowski product and the coproduct ∆ defined in Eq. ( 22). We observe that the idempotence (C, Λ C ) • (C, Λ C ) = (C, Λ C ) hinders the compatibility between the product and the coproduct. For example, taking (C, Λ C ) = ( e 1 , Ze 1 ), then ∆(C • C) = ∆(C) • ∆(C) does not hold. However, the following result shows that this compatibility can be recovered in the context of locality bialgebras.

Proposition 5.2. (QC, ⊥ Q , •, u, ∆, ε) is a connected locality bialgebra.

Proof. We first verify the compatibility of the locality coalgebra structure and the locality algebra structure. To show that ∆ is a locality algebra homomorphism, we note that for (C, Λ C ) ⊥ Q (D, Λ D ), the faces of (C, Λ C ) • (D, Λ D ) are of the form (F 1 , Λ F 1 ) • (F 2 , Λ F 2 ) with F 1 a face of C and F 2 a face of D, and

t((C, Λ C ) • (D, Λ D ), (F 1 , Λ F 1 ) • (F 2 , Λ F 2 )) = t((C, Λ C ), (F 1 , Λ F 1 )) • t((D, Λ D ), (F 2 , Λ F 2 )).
So by definition, we have the desired equation: The counit ε : QC → Q is evidently an algebra homomorphism and hence a locality algebra homomorphism.

Finally, we have checked that QC is both a graded locality algebra (Lemma 3.18) and a connected locality coalgebra (Lemma 4.5). This completes the proof. Definition 5.3. A locality Hopf algebra is a locality bialgebra (B, ⊤, m, ∆, u, ε) with an antipode, defined to be a linear map S : B → B such that S and Id B are mutually independent (in the sense of Definition 3.13) and S ⋆ Id = Id ⋆ S = uε.

The usual proof (see e.g. [START_REF] Guo | An introduction to Rota-Baxter algebra[END_REF]18]) for the existence of the antipode on connected bialgebras extends to locality bialgebras as follows. For k ≥ 1, denote m 1 = m and m k = m(Id B ⊗ m k-1 ). Lemma 5.4. Let (B, ⊤, m, u, ∆, ε) be a connected locality bialgebra, ∆⊗k as in Eq. ( 24) and α : B → B a locality linear map with α(1 B ) = 0. Then (i) α ⋆k = m k-1 α ⊗k ∆(k-1) for all k ≥ 2; (ii) α ⋆k (B n ) = {0} for all k ≥ n + 1. where we have used the locality property of α and the fact that m is associative. (ii) is a direct consequence of (i) and Lemma 4.7.(iv).

  and Λ C is a lattice in the linear subspace spanned by C. Let C k be the set of lattice cones in R k and C = k≥1 C k be the set of lattice cones in (R ∞ , Z ∞ ). Let QC k and QC be the linear spans of C k and C over Q.

  is a locality semi-group in the sense of Definition 3.1.(i). (ii) An locality algebra is a nonunitary locality algebra (A, ⊤, m A ) together with a unit 1 A : K → A in the sense that (A, ⊤, m A , 1 A ) is a locality monoid defined in Definition 3.1. (iii). We shall omit explicitly mentioning the unit 1 A and the product m A unless this generates an ambiguity. (iii) A linear subspace B of a locality algebra (A, ⊤, m A ) is called a sub-locality algebra of A if B is a sub-locality semigroup of A in the sense of Definition 3.1.(v). (iv) A sub-locality algebra I of a locality commutative algebra (A, ⊤,m A ) is called a locality ideal of A if for any b ∈ I we have b ⊤ • b ⊆ I for all b ⊤ ∈ {b} ⊤ . (v) An locality-linear map f : (A, ⊤ A , • A ) → (B, ⊤ B , • B ) between two (non necessarilyunital) locality algebras is called a locality algebra homomorphism if[START_REF] Guo | An introduction to Rota-Baxter algebra[END_REF]

( 16 )

 16 P (a) P (b) = P (P (a) b) + P (a P (b)) + λ P (a b) for all (a, b) ∈ ⊤.

  ) ⊗ c 2,(2) with c 2,(1) ⊤c 2,(1) and {c 2,(1) , c 2,(2) } ⊆ {c 1 } ⊤ . Thus c 1 ∈ {c 2,(1) , c 2,(2) } ⊤ . By Eq. (20) again, ∆(c 1 ) = (c 1 ) c 1,(1) ⊗ c 2,(2) with c 1,(1) ⊤c 1,(2) and {c 1,(1) , c 1,(2) } ⊆ {c 2,(1) , c 2,(2) } ⊤ . This shows that (c 1,(1) , c 1,(2) , c 2,(1) , c 2,(2) ) is in C ⊤ 4 and hence (c 1,(1) ⊗ c 1,(2) ), (c 2,(1

4. 4 .

 4 The Locality Conservation Principle. We now put together all the locality structures we have obtained so far to provide an answer to the question addressed by the Locality Conservation Principle proposed in Problem 1.3. It is formulated as the preservation of locality by the algebraic Birkhoff factorisation in the coalgebra context in [15, Theorem 4.4]. See Section 5.3 for the formulation of the Locality Product Conservation Principle built on the algebraic Birkhoff factorisation in the Hopf algebra context originated from Connes-Kreimer[START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF].Theorem 4.10. (Algebraic Birkhoff factorisation, locality coalgebra version) Let (C, ⊤ C , ∆) be a connected locality coalgebra, C = ⊕ n≥0 C n , C 0 = KJ. Let (A, ⊤ A , •) be a commutative locality algebra with decomposition A = A 1 ⊕A 2 as a vector space satisfying the following (Basic Assumption) the linear projections π i onto A i along A î, { î} := [2]\{i}, i = 1, 2, are independent locality linear maps and 1 A is in A 1 . Let φ : (C, ⊤ C ) -→ (A, ⊤ A ) be a locality linear map such that φ(J) = 1 A . Then there are unique independent locality linear maps φ

1 :

 1 The map is denoted by S o in[15] where it is shown, by the algebraic Birkhoff factorisation for connected coalgebras [15, Theorem 4.4], that bothS ⋆(-1) QC → M Q,+ and S 2 : QC → M Q,-are linear maps sending the trivial lattice cone ({0}, {0}) to 1 in M Q . Applying Theorem 4.10, we further learn that both S ⋆(-1) 1 and S 2 are locality linear maps. This means that for cones (C, Λ C ) and (D, Λ D ) with (C, Λ C ) ⊥ Q (D, Λ D ), we also have

5. 1 .

 1 Locality bialgebras and locality Hopf algebras. Definition 5.1.

  C, Λ C ) • (D, Λ D )) = ∆(C, Λ C ) • ∆(D, Λ D ).

  Proof. (i) We proceed by induction on k ≥ 2 and first observe that α ⋆k (1 B ) = 0 for every k ≥ 1 as can easily be shown by induction. The result holds for k = 2 since α(1 B ) = 0. Assuming it holds at degree k we writeα ⋆k+1 (x) = m(α ⊗ α ⋆k )∆(x) = m(α ⊗ α ⋆k ) ∆(x) (since α ⋆k (1 B ) = α(1 B ) = 0) = m α ⊗ (m k-1 α ⊗k ∆(k-1) ) ∆(x) = m(Id B ⊗ m k-1 )(α ⊗ α ⊗k )(Id B ⊗ ∆(k-1) ) ∆(x) = m k α ⊗(k+1) ∆(k) (x)

Proposition 5 . 5 .

 55 Let (B, ⊤, m, u, ∆, ε) be a connected locality bialgebra. There is a linear map S : B → B with the properties of the antipode stated above. It is given byS = ∞ k=0 (uε -Id) ⋆k .Proof. The map α : B → B defined by α = Iduε is locality linear, and α(1 B ) = 0. The von Neumann series S = ∞ k=0 (-1) k α ⋆k which is locally finite by Lemma 5.4.(ii) and hence well-defined, gives the inverse of the identity for the convolution product.As an immediate consequence of Propositions 5.2 and 5.5, we have Corollary 5.6. The locality bialgebra (QC, ⊥ Q , •, u, ∆, ε) is a locality Hopf algebra.5.2. Locality of the convolution of locality algebra homomorphisms.Proposition 5.7. Let (B, ⊤ B , m, u, ∆, ε) be a locality bialgebra. Let (A, ⊤ A , •) be a locality commutative algebra. Let φ, ψ : (B, ⊤ B ) -→ (A, ⊤ A ) be independent locality linear maps. (i) If φ and ψ are locality multiplicative then so is their convolution product φ ⋆ ψ.(ii) Assume further that B is connected. If φ is a homomorphism of locality algebras, then so is its convolution inverse φ ⋆(-1) . So the set G of homomorphisms of locality algebras from (B, ⊤ B ) to (A, ⊤ A ) is a locality group with respect to the independent relation of locality linear maps.Proof. (i) For (c, d) ∈ ⊤ B , by the proof of Lemma 4.6.(iii), we can write∆(c) = i c i1 ⊗ c i2 , ∆(d) = j d j1 ⊗ d j2 with (c i1 , c i2 , d j1 , d j2 ) ∈ B ⊤ 4 . Then ∆(cd) = (m ⊗ m)τ 23 (∆ ⊗ ∆)(c ⊗ d) = i,j c i1 d j1 ⊗ c i2 d j2 . So (φ ⋆ ψ)(c d) = i,j φ (c i1 d j1 ) ψ (c i2 d j2 ) = i,j φ (c i1 ) φ (d j1 ) ψ (c i2 ) ψ (d j2 ) = i φ (c i1 ) ψ (c i2 ) j φ (d j1 ) ψ (d j2 ) = (φ ⋆ ψ(c)) (φ ⋆ ψ(d)) .(ii) Now we use induction on the sum of degrees of c and d, c⊤d to proveφ ⋆(-1) (c)φ ⋆(-1) (d) = φ ⋆(-1) (cd),which is true if the sum of degrees is 0. In general, by Lemma 4.7.(ii), we write∆(c) = c ⊗ J + J ⊗ c + (c) c ′ ⊗ c ′′ , ∆(d) = d ⊗ J + J ⊗ d + (d) d ′ ⊗ d ′′ with (c ′ , c ′′ , d ′ , d ′′ ) ∈ B ⊤ 4 . So by ∆(cd) = ∆(c)∆(d), we obtain ∆(cd) = cd ⊗ J + J ⊗ cd + c ⊗ d + d ⊗ c + (d) cd ′ ⊗ d ′′ + (d) d ′ ⊗ cd ′′ + (c) c ′ d ⊗ c ′′ + (c) c ′ ⊗ c ′′ d + (c)(d) c ′ d ′ ⊗ c ′′ d ′′ .By Eq. (26) we obtainφ ⋆(-1) (cd) = -φ(cd)φ(c)φ ⋆(-1) (d)φ(d)φ ⋆(-1) (c) -(d) φ(cd ′ )φ ⋆(-1) (d ′′ ) -(d) φ(d ′ )φ ⋆(-1) (cd ′′ ) -(c) φ(c ′ d)φ ⋆(-1) (c ′′ ) -(c) φ(c ′ )φ ⋆(-1) (c ′′ d) -(c)(d)φ(c ′ d ′ )φ ⋆(-1) (c ′′ d ′′ ).By Eq. (26) applied to c and d, the locality multiplicativity of φ, the commutativity of A and induction hypothesis, we have φ ⋆(-1) (cd) = φ(c)φ(d) +

  )φ(d ′ )φ ⋆(-1) (d ′′ ) + (c) φ(c ′ )φ(d)φ ⋆(-1) (c ′′ ) + (c)(d) φ(d ′ )φ ⋆(-1) (d ′′ )φ(c ′ )φ ⋆(-1) (c ′′ ) + (c)(d) φ(c ′ )φ ⋆(-1) (c ′′ )φ(d ′ )φ ⋆(-1) (d ′′ ) -(c)(d) φ(c ′ d ′ )φ ⋆(-1) (c ′′ d ′′ ) = φ(c)φ(d) + (d) φ(c)φ(d ′ )φ ⋆(-1) (d ′′ ) + (c) φ(c ′ )φ(d)φ ⋆(-1) (c ′′ )

  1, one can hope for a partially defined multiplication preserved by renormalisation. So we propose the following Problem 1.4. (Locality Product Conservation Principle in Renormalisation) Consider a connected bialgebra H and a unital commutative algebra A with a linear map P as in Theorem 1.1. Let ⊤ H ⊆ H × H and ⊤ A ⊆ A × A be relations on H and A respectively for which a partial multiplication m H : ⊤ H → H and m A : ⊤ A → A are defined. Let φ : H → A be partially multiplicative in the sense that φ(m H

  The set G of linear subspaces of R 2 is a locality set with respect to the following relation ⊤ G on linear subspaces of G: U, V ⊆ R 2 are called transverse if they intersect trivially, namely if U ∩ V = {0}. The set G equipped with linear sums + is a monoid. But the corresponding (G, ⊤ G , +) is not a locality monoid. Indeed, for the standard basis {e 1 , e 2 } of R 2 , the subspaces Re 1 and Re 2 both intersect R(e 1 + e 2 ) trivially, but Re 1 + Re 2 does not. The locality set (M Q , ⊥ Q ) in Definition 2.8, equipped with the restricted multiplication m

	Example 3.4.

y 1 , y 2 ) and (x 1 , x 2 , y 1 y 2 ) are in G ⊤ 3 and hence (x 1 x 2 , y 1 y 2 ) ∈ ⊤. As a simple counter example of locality semigroup, we have Counterexample 3.3.

As a special case of partial algebras[12], the terminology "partial semigroup" is used for a set equipped with a partial associative product defined only for certain pairs of elements in the set. The condition for a locality semigroup is more restrictive than that of a partial semigroup in that the former requires that the pairs for which the partial product is defined stem from a symmetric relation and that the partial product should be compatible with the locality relation in the sense of Eq. (9).

Acknowledgements. The authors acknowledge supports from the Natural Science Foundation of China (Bin Zhang: Grants No. 11071176, 11221101 and 11521061, Li Guo: Grant No. 11371178) and the German Research Foundation (Sylvie Paycha and Pierre Clavier: DFG grant PA 1686/6-1). They are grateful to the hospitalities of Sichuan University and University of Potsdam where parts of the work were completed.

Let ψ and χ be independent locality linear maps in U ⊤ L and let φ be in U. Then φ, ψ, χ are pairwise independent. Therefore φ × ψ × χ : C ⊤ 3 -→ A ⊤ 3 is well defined. For (c 1 , c 2 ) ∈ C ⊤ 2 , that is c 1 ∈ {c 2 } ⊤ , there exist (d 1 , e 1 ),

with (d i , e i , c 2 ) ∈ C ⊤ 3 . Then (ψ(d i ), χ(e i ), φ(c 2 )) ∈ A ⊤ 3 and hence (ψ(d i )χ(e i ))⊤ A φ(c 2 ). So we have (ψ ⋆ χ)(c 1 )⊤ A φ(c 2 ), which means ψ ⋆ χ is in φ ⊤ L . Thus ψ ⋆ χ ∈ U ⊤ L . This verifies the first axiom. The associativity of ⋆ follows from the associativity of m and coassociativity of ∆ as in the classical case.

(ii) We now assume that C is a connected locality coalgebra. For a locality linear map φ : C → A, we now prove by induction on the degree of c 1 that the map

is well defined, and that

This is trivial for degree 0 since φ ⋆(-1) (J) = 1 A . Assume for any c 1 ∈ C of degree ≤ n, φ ⋆(-1) (c 1 ) is well defined, and for c with

By the induction hypothesis, φ ⋆(-1) (c ′′ 1 ) is well defined, such that φ ⋆(-1) (c ′′ 1 )⊤ A φ(c) and φ ⋆(-1) (c ′′ 1 )⊤ A φ(c ′ 1 ). Since φ is a locality linear map, we also have φ(c 1 )⊤ A φ(c) and φ(c ′ 1 )⊤ A φ(c). Thus φ(c ′ 1 )φ ⋆(-1) (c ′′ 1 ) is well defined and (φ(c ′ 1 )φ ⋆(-1) (c ′′ 1 ))⊤ A φ(c). So, φ ⋆(-1) (c 1 ) is well defined and φ ⋆(-1) (c 1 )⊤ A φ(c), which means φ⊤ L φ ⋆ (-1) .

Again by induction on the degree of c 1 , we now prove that φ ⋆(-1) is a locality linear map by checking

a fact which is obvious at degree 0. Assume that, for a given n ≥ 0 and any c 1 of degree ≤ n the equation holds. Consider c

. Therefore, we conclude that GL is a locality group with unit u A ε C by Example 4.8.

(iii) Similar inductions show that φ⊤ L ψ implies φ⊤ L ψ ⋆(-1) and φ ⋆(-1) ⊤ L ψ ⋆(-1) , so that φ ⋆k ⊤ L ψ ⋆l for any k, l ∈ Z. is well-defined, and for any d ∈ C with d⊤ C c, there is

which clearly hold for c of degree 0.

Assume that these hold true for c of degree ≤ n. Then for c of degree n + 1, c ′ is of degree ≤ n, so φ 1 (c ′ ) is defined and φ 1 (c ′ )⊤ A φ(c ′′ ). Therefore φ 1 (c ′ )φ(c ′′ ) makes sense, and φ 1 (c) is well-defined.

Further for any d ∈ C with c⊤ C d, we can take {c ′ , c ′′ }⊤ C d. Since φ is a locality map, we obtain φ(c)⊤ A φ(d) and φ(c ′′ )⊤ A φ(d) . Also the induction hypothesis gives

Now since π 1 is a locality map and π 1 and π 2 are independent, π 1 and Id A = π 1 + π 2 are independent. Thus φ 1 (c)⊤ A φ(d). Therefore we have proved that φ 1 is well-defined and

. By a similar induction on the degree of c, we obtain φ 1 (c)⊤ A φ 1 (d), so φ 1 is a locality linear map. Therefore, the map

Notice that for c ∈ C n , n > 0, Eq. ( 31) means

With the condition on J, this in turn reads φ 2 = φ 1 ⋆ φ and hence φ = φ ⋆(-1) 1

⋆ φ 2 . By Proposition 4.9, φ ⋆(-1) 1 is a locality linear map. From Eq. (32), we easily obtain φ 1 ⊤ L φ 2 . By Eq. ( 26), an easy induction on the degree of c shows φ

A similar induction shows that if ψ : (C, ⊤ C ) -→ (A, ⊤ A ) is also a locality map, independent of φ with φ(J) = 1 A , then φ i and ψ j are independent for i, j = 1, 2, proving the last statement of the theorem. Now to prove (i), letting A 1 be a sub-locality algebra, Eq. ( 26) and a simple induction on n ≥ 0 show that φ

To prove (ii), suppose A 1 is a sub-locality algebra and A 2 is a locality ideal. We prove by induction on n ≥ 0 that

so Eq. (33) holds for n = 0 since C 0 = KJ. Assuming that Eq. (33) holds for any c in C of degree ≤ n, we prove that Eq. (33) holds for any element c ∈ C n+1 .

We write

This completes the induction.

be a locality algebra homomorphism. Then there are unique independent locality algebra homomorphisms φ i :

The map φ

is also a locality algebra homomorphism and

with ∆(c) = (c) c ′ ⊗ c ′′ the reduced coproduct defined in Lemma 4.7. Remark 5.9. Theorem 5.8 provides an answer to Problem 1.4: the locality multiplicativity of the renormalised map follows from that of the initial map under the assumption of the theorem.

Proof. All the statements follows from Theorem 4.10 except the claims that φ i , i = 1, 2 and φ ⋆(-1) 1 are locality algebra homomorphisms. For c⊤ H d, by Lemma 4.7.(ii), we can write

. Using the locality Rota-Baxter property of P , by a similar argument as in the non-locality case [START_REF] Guo | An introduction to Rota-Baxter algebra[END_REF]Theorem 2.4.3], we can prove that φ 1 and φ 2 are homomorphisms of locality algebras. By Proposition 5.7, φ ⋆(-1) 1 is a homomorphism of locality algebras.

Applying Theorem 5.8 to (A, ⊤ A , •) = M Q , ⊥ Q , • yields the following result.

Corollary 5.10. Let (H, ⊤ H ) be a connected locality Hopf algebra. Let

(iv) assuming in addition that φ is a locality algebra homomorphism, then the maps π Q 1 φ, φ Q 1 and φ Q 2 are locality algebra homomorphisms.

Proof. The proof is straightforward; let us nevertheless mention that π Q 1 φ⊤ L φ Q 2 follows from φ 1 ⊤ L φ 2 in Eq. ( 36) combined with the fact that the convolution inverse preserves locality.

We end the paper by applying Corollary 5.10 to the locality algebra homomorphism S : QC → M Q . The classical algebraic Birkhoff factorisation does not apply while Theorem 4.10 does, telling us that S ⋆(-1) 1

: QC → M Q,+ and S 2 : QC → M Q,- are locality linear maps. From Corollary 5.10, we conclude that the two maps are also locality algebra homomorphisms and thus are multiplicative for orthogonal pairs of lattice cones. Noting further [15] that S 2 = I, the exponential integral and S Further, µ = π Q + S.


2.5D Evidential Grids for Dynamic Object Detection

Hind Laghmara, Thomas Laurain, Christophe Cudel and Jean-Philippe Lauffenburger IRIMAS EA7499, Université de Haute-Alsace Mulhouse, France firstname.name@uha.fr Abstract-Perception is a crucial and challenging part of Intelligent Transportation Systems. One of the main issues is to keep up with the moving objects in complex and dynamic environments. This paper proposes a method for dynamic object detection using Evidential 2.5D Occupancy Grids. The approach is based on a map representation for occupation modeling and navigable area definition. At each time step, a local grid is derived from the sensor data. Belief Theory is then retained to perform a grid fusion over time in order to keep track of the moving objects in the grid. The description of the dynamic behavior of objects in a scene is related to the conflict issued after the temporal fusion. Finally, the construction of the objects themselves is realized with a segmentation based on Density-Based Spatial Clustering of Applications with Noise (DBSCAN). In order to validate the efficiency of the proposed approach, experimental results are provided based on the KITTI dataset. Performances are evaluated through comparison with the ground truth.

Index Terms-Occupancy Grids, Belief Theory, Dynamic Object Detection, Autonomous Vehicles, LiDAR.

I. INTRODUCTION

This paper focuses on the perception of Intelligent Transportation Systems for which the objective is to model the vehicle's local environment according to data issued from multiple sensors. This paper aims to achieve such task considering that the vehicle's pose is known. The surrounding environment representation is based on Occupancy Grids (OG) as it indicates two main features: the navigable space as well as the location of obstacles which may be static or dynamic. OGs can be constructed based on the measures of the objects' distance from the ego-vehicle which can be given by exteroceptive sensors such as LiDARs (Light Detection And Ranging), Radars or stereo vision.

The main challenges considering this issue are the uncertainty and imprecision of information as well as the complexity that lies behind modeling a dynamic scene. Multiple Object Tracking (MOT) is the application allowing to handle this issue which includes the detection, association and tracking of dynamic objects sequentially. OGs have proven to be effective with a limited computation complexity for local environment modelling taking account of temporal data. The aim in this paper is to improve OGs to insure a robust detection of dynamic objects. Due to the high uncertainty in such task, the use of Dempster-Shafer theory is convenient for autonomous systems' applications. Some of the main references in the literature to treat the detection of dynamic objects with grid-based solutions are reviewed in Section II-A. A survey is also presented in [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF] where a 2.5D (two dimensions plus the elevation information obtained by averaging the height of all points that fall into a given cell) approach is used for the determination of moving cells.

This paper will be based on the same representation to include a tri-dimensional modeling of the environment. The main contribution of this approach is that the classification of dynamic objects in a 2.5D grid is done according to an evidential fusion of multi-grids. The second aim was to extract an object-level representation according to the detected dynamic cells for tracking purposes.The objective is to achieve the dynamic detections on an object-level rather than on a cell-level which is commonly done in literature. The objects are built based on the clustering of mobile cells using the DBSCAN algorithm. The third contribution of this work is a quantitative evaluation according to a measure of average precision of the detection results based on a KITTI dataset for comparison purposes.

The paper is structured as follows: Section II covers a survey on multiple object detection based on occupancy grids as well as the definition of a 2.5D representation. Section III introduces the different steps allowing the transition from a grid-level representation to the dynamic object detection. This includes the definition of an evidential grid as well as mobile cells labeling (cf. Fig. 1). The segmentation algorithm is also specified in order to extract dynamic objects. Section IV illustrates the dataset used for evaluation as well as a quantitative result analysis. Section V concludes the paper.

II. 2.5D GRID MAPS A. Related Work

An OG is a representation which employs a multidimensional tesselation of the space into cells where each one stores a knowledge of its state of occupancy [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF]. Today, there is a large use of OGs due to the availability of powerful resources to handle their computational cost. The construction of a grid has been applied in multiple dimensions (2D, 2.5D and 3D) [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF] using different sensor technologies like 2D radars, 2D or 3D LiDARs and stereo-vision. In this representation, each cell-state is described according to a chosen formalism. The most common one is the Bayesian framework which was adopted first by Elfes [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF] followed by many extensions as the well-known Bayesian Occupancy Filter (BOF) [START_REF] Coué | Bayesian Occupancy Filtering for Multitarget Tracking: an Automotive Application[END_REF][START_REF] Broggi | A full-3d voxel-based dynamic obstacle detection for urban scenario using stereo vision[END_REF]. The latter estimates the dynamics of the grid cells using the Fast Clustering and Tracking Algorithm in order to ensure MOT [START_REF] Mekhnacha | The "fast clustering-tracking" algorithm in the bayesian occupancy filter framework[END_REF].

Other works suggested a formalism based on Dempster-Shafer (or Evidence) Theory. It has been applied in [START_REF] Pagac | An evidential approach to map-building for autonomous vehicles[END_REF] based on a 2D occupancy grid using a ring of ultrasonic transducers and a sensor scanner. Moras et al. proposed a similar approach also used for mobile object detection based on an inverse sensor model [START_REF] Moras | Moving objects detection by conflict analysis in evidential grids[END_REF][START_REF] Moras | Credibilist occupancy grids for vehicle perception in dynamic environments[END_REF]. The latter is realized according to the conflict analysis by a temporal evidential fusion of multiple grids. Extending Moras et al.'s work, contextual discounting is applied in [START_REF] Kurdej | Controlling remanence in evidential grids using geodata for dynamic scene perception[END_REF] to control cell remanence.

Some references study the dynamics of the environment at the cell level to avoid the inconsistencies of the object representation [START_REF] Danescu | Modeling and tracking the driving environment with a particlebased occupancy grid[END_REF]. Tanzmeister et al. [START_REF] Tanzmeister | Evidential grid-based tracking and mapping[END_REF] also estimate the static and dynamic characteristics at the grid cell level and use a particle filter for obtaining the cell velocity distribution. Honer et al. [START_REF] Honer | Motion state classification for automotive lidar based on evidential grid maps and transferable belief model[END_REF] focus on the classification of stationary and dynamic elements based on an evidential semantic 2D grid map with a five state cell configuration: each cell can either be free, dynamic, static, occupied or unknown. However, the update of cells is done according to a combination table heuristically determined.

The above literature review and especially [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF] show that most of the works consider a two-dimensional grid for the environment representation even when 3D sensors are providing the data to build the map. In fact, 3D solutions like voxel grids or octomaps can generate high complexity and computation load when applied to real-time applications like autonomous navigation. An interesting tradeoff remains in 2.5D occupancy grid which are known to be memory efficient and at the same time store elevation data. In the particular context of autonomous driving and ITS, in which the elevation variation of the terrain is limited in the local area in which the vehicles are driving, 2.5D representations are of real interest and are retained here.

In this work, the objective is to consider an object-oriented tracking which necessitates an efficient object detection module. The idea is to consider the tri-dimensional sensor data issued by a Velodyne LiDAR to built at each time step a 2.5D grid for which the elevation is attributed to each cell. Sections II-B and III describe the approach illustrated in Fig. 1.

B. Building a 2.5D Grid

The pre-processing step from Fig. 1 required to build a 2.5D grid is derived from [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF]. The grid is composed of discrete cells in which the object height above the ground level is stored. This representation can describe the elevated objects from the ground which can correspond to dynamic or static objects.

Building the 2.5D grid includes defining the covering area as well as its resolution, which corresponds to the dimensions of each cell. An example of a 2.5D grid map is shown in Fig. 2 where the resolution is 0.4 × 0.4m. The grid covers 40m in front, 20m behind and 20m along right and left sides of the vehicle. In order to consider the elevation of objects, it is necessary to determine all measures that correspond to the ground. Several approaches as [START_REF] Wang | Lidar ground filtering algorithm for urban areas using scan line based segmentation[END_REF] treat this point because it can induce errors when investigating the occupancy. Such cases are very frequent when the road is uneven or tilted. In this work, the method presented in [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF] is employed. It consists in evaluating the variance of height of the points which correspond to a cell. If this value is larger than a threshold, the average height is verified to surpass a defined value in order to make sure it belongs to the ground surface and not any other planar surface. This is equivalent to the following statement:

G(i, j) = 0 if σ 2 i,j < tr σ and µ i,j < tr µ µ i,j otherwise , (1)
where µ i,j and σ 2 i,j are the average height and its variance in cell with index (i, j). The thresholds tr σ and tr µ defined in [START_REF] Asvadi | Detection and tracking of moving objects using 2.5d motion grids[END_REF] are respectively equal to 2cm and 30cm. G is the resulting 2.5D grid which will be further used for object detection.

III. FROM AN EVIDENTIAL GRID TO THE OBJECT LEVEL A. Modeling an Evidential Grid

Extending the probability theory, the Belief Theory offers an adequate representation of the data and source imperfections and thus is appropriate for perception in ITS. It offers a wide range of fusion operators handling these properties according to the application.

In this work, the solution from [START_REF] Moras | Moving objects detection by conflict analysis in evidential grids[END_REF] is adapted to the 2.5D grid. Moras et al. suggest an approach based on the conflict appearing during the temporal grid fusion for mobile object detection and navigable space determination. For that, a frame of discernment is defined to include the states of a cell considering it to be Free (F) or Occupied (O). The discernment frame is then Ω = {F, O}.

The referential power set contains all possible combinations of the discernment frame hypotheses:

2 Ω = {∅, F, O, {F, O}}.
To express the belief in each state, a mass function m(.) is defined to respectively express conflict m(∅), Free state m(F ), Occupied state m(O) and the unknown state m({F, O}).

B. Inverse Sensor Model

Basically, a sensor model is how the mass function of a state according to a measure is calculated. This basic belief assignment (bba) also includes the reliability of the source. In this application, the considered sensor is a 3D multi-echo LiDAR provided by Velodyne. The input data will include ranges r i and angles θ i of each laser beam or point p as shown in Fig. 3.

According to this set of data, a Scan Grid (SG) in polar coordinates is constructed. Each row of this SG corresponds to an angular sector Θ = [θ -, θ + ] for which a cell is defined in R × Θ. The range of a cell is R = [r -, r + ] which means that each cell is defined by a pair on which a mass is attributed as m{Θ, R}. The masses corresponding to each proposition A ∈ Ω are found hereby [START_REF] Moras | Moving objects detection by conflict analysis in evidential grids[END_REF]:

m{Θ, R}(∅) = 0 (2) m{Θ, R}(O) = 1 -µ F r i ∈ R 0 otherwise (3) m{Θ, R}(F ) = 1 -µ O if r + < min(r i ) 0 otherwise (4) m{Θ, R}(Ω) =      µ F r i ∈ R µ O if r + < min(r i ) 1 otherwise (5)
where µ F and µ O respectively correspond to the probability of false alarm and missed-detection of the sensor. For simplicity reason, these mass functions will be noted m(∅), m(O), m(F ) and m(Ω).

C. Combination of Evidential Grids

The construction of a SG is sequentially done to translate the sensor's data. However, the temporal propagation of the knowledge and uncertainties provided by every point cloud given by the sensor requires a fusion process between the current SG and the result of the previous fusion. The complete description of the environment resulting from such a combination provides a Map Grid (MG). This update allows to detect the consistencies of data as well as some cases of conflict. Fig. 4 illustrates the process of building and updating a MG using the sensor point cloud provided at a time t. It is the outcome of a combination of a SG built at t according to (2) and a transformed MG built at t -1.

The grid transformation is applied with respect to the new pose of the vehicle at t in order to guarantee that the information is expressed in the current coordinate system of the vehicle. This operation is realized by a spatial transformation for which the aim is to associate to each cell new coordinates. Algorithm 1 describes the approach. Initialize M G t-1,tr cells with m M Gt-1,tr (Ω) = 1 for each cell with index (p, q) do Apply a change of coordinates (p, q) = R × (p, q) + T Calculate the new indices

(p new , q new ) = min(|ceil(p, q)|, |f loor(p, q)|) × sign(i, j) if (p new × q new ) > 0 then M G t-1,tr (p new , q new ) = M G t-1 (p, q) end if end for
This update is done according to an evidential multi-grid fusion. This is the crucial point of the grid-based object detection process as it allows the temporal update of the map grid and also the evaluation of the state of cells. Among the various operators in Belief Theory, Dempster's rule of combination is used:

m M Gt = m M Gt-1,tr ⊕ m SGt ( 6
)
where m M Gt-1,tr and m SGt are resp. the mass function of the transformed MG and SG at time t. The operator is defined as:

(m 1 ⊕m 2 )(A) = K ∀B,C∈2 Θ ,B∩C=A,A =∅ m 1 (B).m 2 (C) (7)
where

K -1 = 1 - ∀B,C∈2 Θ ,B∩C=∅ m 1 (B).m 2 (C) (8)
The resulting masses m M Gt (A) define the state of each cell which depends on the previous state and the new measures. The resulting masses according to each state are found as follows [START_REF] Moras | Moving objects detection by conflict analysis in evidential grids[END_REF]:

m M Gt (O) =m SGt (O)m M Gt-1,tr (O) + m SGt (Ω) m M Gt-1,tr (O) + m SGt (O)m M Gt-1,tr (Ω) m M Gt (F ) =m SGt (F )m M Gt-1,tr (F ) + m SGt (Ω) m M Gt-1,tr (F ) + m SGt (Ω)m M Gt-1,tr (F ) m M Gt (Ω) =m SGt (Ω)m M Gt-1,tr (Ω) m M Gt (∅) =m SGt (O)m M Gt-1,tr (F ) + m SGt (F ) m M Gt-1,tr (O) (9)
with m M Gt (∅) being the combined mass expressing the conflict. Basically, this property shows the discordance between the knowledge expressed at t -1 and t. The reason for which a conflict appears is that when a cell changes its state from F to O or vice-versa. Therefore, the detection of this conflict can lead to the evaluation of the dynamic cells. The conflict allows to label the occupied cells which change their state according to two types of conflict:

C 1 =m SGt (O).m M Gt-1,tr (F ) from F to O C 2 =m SGt (F ).m M Gt-1,tr (O) from O to F ( 10
)
where

m M Gt (∅) = C 1 + C 2 .
Dempster's operator implies a normalization of conflict at fusion considering its absorbing property. Basically, if the conflict is included in the next combination, it induces loss of information because m M Gt (∅) increases at each fusion. Therefore, the updated grid contains no conflict. It is only preserved to classify the mobile cells to be studied for dynamic object extraction.

D. Clustering for Dynamic Object Detection

In order to attain the object level representation from the mobile cells, a clustering is applied to group those cells related to the same object in the grid. For that, the partitioning method must be unsupervised considering that the number of objects to be found is unknown. The well-known Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is used [START_REF] Ester | A densitybased algorithm for discovering clusters in large spatial databases with noise[END_REF]. It is based on the estimated density of measures for partitioning clusters. This algorithm uses two main parameters: the minimal distance and the minimum number of points minP ts which must reside within a radius to be included in a cluster. This algorithm is convenient because it is simple and can handle aberrant or noisy values while clustering. However, it can have some issues when clusters have different densities. The clustering algorithm is applied to a set of cells which should necessarily be occupied (i.e. non-zero elevation) and for which the conflict m M Gt (∅) is later used to classify the set of the resulting clusters. The partial conflict C 2 informs about cells changing state from occupied to free at a time t. The cells affected with this conflict do not belong to a given object and hence do not provide any knowledge about the object's presence. That is why, it is trivial to only consider the partial conflict C 1 to determine the location of the object at time t. However, exclusively clustering the C 1 -labeled cells is not informative enough to obtain a complete representation of the shape of the dynamic object. On the grid, the displacement of objects is only visible at the perimeter, the conflict is mostly located at the boundaries of objects as shown in Fig. 5. For that, clustering is applied to detect both classes of static and dynamic objects according to the elevation measure on the 2.5D grid. Afterwards, a classification of these clusters is made according to those which partially contain conflictual cells as mentioned in Fig. 1.

IV. EXPERIMENTAL RESULTS

The presented approach is applied to real data and has been tested offline. The validation is done at the grid-level as well as the object-level according to the ground truth (GT) for qualitative and quantitative evaluation.

A. Dataset and Performance Evaluation

The used data is extracted from the KITTI database [START_REF] Geiger | Vision meets robotics: The kitti dataset[END_REF]. It is a widely used dataset in research for autonomous driving as it provides a large set of images, GPS/IMU recordings and laser scans raw data as well as labeled scenes. In this study, sequence 17 from the raw data set is used considering that annotations with the GT on detected objects as well as the vehicle pose are available. A total of 114 frames are used for which 59 contain annotated moving cars. The used data for this approach are the point cloud recorded by a Velodyne HDL-64 which is characterized by 64 horizontal layers and a 360 o horizontal field of view and a 26.9 o vertical field of view. The GPS data is also used to obtain the vehicle's pose. The images are not exploited for this application but are used for visualization.

3D object detection benchmarks offer various criteria for performance evaluation purposes. The most common measure is the precision. It is the proportion of all example above the rank which are from the positive class [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF]:

P recision = T P T P + F P (11)
where T P and F P respectively stand for True Positive and False Positives. This metric is calculated according to the overlap of candidate detections with the GT. For the computation, the bounding box of a detected object is compared to the GT bounding box. A correct detection has an overlap area a o which exceeds 50% to be considered True and can be calculated as follows:

a o = area(B p ∩ B gt ) area(B p ∪ B gt ) (12)
where B p and B gt are respectively the candidate bounding box and the GT bounding box.

B. Results

In the following section, the perception results will be illustrated according to the 2.5D grid, the evidential occupancy measures and the detected objects. The results are confronted with the available GT-based objects.

Fig. 6 shows an example of results according to the image captured by the camera facing the front of the vehicle. The corresponding 2.5D grid found below the image expresses the average height of elevated objects in the scene. In this view, the car's position is approximately x = 50, y = 50 and heading to the right. It can been seen that this map contains voxels describing moving objects (2 cars) as well as static ones like numerous traffic signs or static vehicles behind the ego-car. The grid fusion allows to determine which among these voxels belong to dynamic objects. The bottom figure, exclusively, shows the results of dynamic objects found in the range of the camera view. It provides a comparison between the bounding boxes resulting from the fusion and clustering process and the GT bounding boxes. We choose not to display the objects detected behind the ego-vehicle since no annotations are available in the GT.

The position of these objects is found according to the evaluation of conflict in the corresponding MG. The conflict C 1 allows to observe the cells changing state from Free to Occupied and the voxels which contain a non-zero value of C 1 are grouped to define the moving objects. Considering that we only detect cars in this sequence, the parameters of DBSCAN are minP ts = 4 and = 5 in grid coordinates. This algorithm is advantageous for this application because it discards the measures which can be considered noisy. This allows to optimize the number of relevant clusters. The extracted clusters are labeled according to bounding boxes containing the exact number of LiDAR data belonging to the cluster. The use of 3D bounding boxes allows to evaluate the results according to the known locations of GT. The total AP is found to be 91.23% with an overlap illustrated in Fig. 7. It can be noticed that most detected objects overlap with true objects at a rate varying between 65% -90%. Note that the number of false alarms is very low due to the property of the clustering algorithm which only considers dense groups of measurement. The noisy or distant data do not belong to any object.

V. CONCLUSION

The approach presented in this paper aims at the detection of multiple objects based on LiDAR data according to an evidential 2.5D grid. The main contributions of this paper is the use of an evidential elevation map and the evaluation of conflict for the determination of mobile objects. Another contribution is the clustering to have an object-level representation for tracking purposes. The detection of dynamic objects is evaluated according to ground truth given by a set of annotations of KITTI dataset. The approach is shown to be efficient according to its high average precision. The perspectives for future work are the identification of the clustering algorithm parameters in order to identify many classes of objects. Extending these first results by testing the approach on more complex scenarios including occluded objects is a second work perspective. Furthermore, a comparative study of this work with the state of the art results will be performed.

Fig. 2 .

 2 Fig. 2. Top: 3D LiDAR point cloud from KITTI, Bottom: Corresponding 2.5D grid with 0.4 × 0.4m cells.

Fig. 1 .Fig. 3 .

 13 Fig. 1. Dynamic object detection with an evidential 2.5D grid.

Fig. 4 .

 4 Fig. 4. Map Grid Construction.

Algorithm 1

 1 Grid Transformation to new vehicle coordinates Require: Previous Map Grid M G t-1 , Rotation matrix R, Translation Vector T . Ensure: Build a transformed Map Grid M G t-1,tr

Fig. 5 .

 5 Fig. 5. Appearance of conflict due to the displacement of objects. Conflict C 1 informs about newly occupied cells whereas C 2 describes transitions between occupied to free.

Fig. 6 .

 6 Fig. 6. Top: Frame 40 of Sequence 17, Middle: The corresponding 2.5D grid, Bottom: Comparison of the detection results on frame 40 with the ground truth.

Fig. 7 .

 7 Fig. 7. Overlap of the detected objects with the ground truth showing the rate above which a detection is eligible.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support from Fondation Wallach (Mulhouse) in the context of the Project SIMPHA (Solution Innovantes pour la Mobilité individualisée et durable des séniors et Personnes présentant un Handicap).