Jean-Louis Giavitto
email: giavitto@lami.univ-evry.fr

Olivier Michel
email: michel]@lami.univ-evry.fr

Julien Cohen

Antoine Spicher

Ada Lovelace

Computations in Space and Space in Computations

Goals and Motivations

The emergence of terms like natural computing, mimetic computing, parallel problem solving from nature, bio-inspired computing, neurocomputing, evolutionary computing, etc., shows the never ending interest of the computer scientists for the use of "natural phenomena" as "problem solving devices" or more generally, as a fruitful source of inspiration to develop new programming paradigms. It is the latter topic which interests us here. The idea of numerical experiment can be reversed and, instead of using computers to simulate a fragment of the real world, the idea is to use (a digital simulation of) the real world to compute. In this perspective, the processes that take place in the real world are the objects of a new calculus: description of the world's laws = program state of the world = data of the program parameters of the description = inputs of the program simulation = the computation This approach can be summarized by the following slogan: "programming in the language of nature" and was present since the very beginning of computer science with names like W. Pitts and W. S. McCulloch (formal neurons, 1943), S. C. Kleene (inspired by the previous for the notion of finite state automata, 1951), J. H. Holland (connectionist model, 1956), J. Von Neumann (cellular automata, 1958), F. Rosenblatt (the perceptron, 1958), etc.

This approach offers many advantages from the teaching, heuristic and technical points of view: it is easier to explain concepts referring to real world processes that are actual examples; the analogy with the nature acts as a powerful source of inspirations; and the studies of natural phenomena by the various scientific disciplines (physics, biology, chemistry...) have elaborated a large body of concepts and tools that can be used to study computations (some concrete examples of this cross fertilization relying on the concept of dynamical system are given in references [START_REF] Chandy | Reasoning about continuous systems[END_REF][START_REF] Brockett | Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems[END_REF][START_REF] Sintzoff | Invariance and contraction by infinite iterations of relations[END_REF][START_REF] Geurts | Hierarchy of discrete-time dynamical systems, a survey[END_REF]).

There is a possible fallacy in this perspective: the description of the nature is not unique and diverse concurent approaches have been developed to account for the same objects. Therefore, there is not a unique "language of nature" prescribing a unique and definitive programming paradigm. However, there is a common concern shared by the various descriptions of nature provided by the scientific disciplines: natural phenomena take place in time and space.

In this paper, we propose the use of spatial notions as structuring relationships in a programming language. Considering space in a computation is hardly new: the use of spatial (and temporal) notions is at the basis of computational complexity of a program; spatial and temporal relationships are also used in the implementation of parallel languages (if two computations occur at the same time, then the two computations must be located at two different places, which is the basic constraint that drives the scheduling and the data distribution problems in parallel programming); the methods for building domains in denotational semantics have also clearly topological roots, but they involve the topology of the set of values, not the topology of a value. In summary, spatial notions have been so far mainly used to describe the running of a program and not as means to design new programs.

We want to stress this last point of view: we are not concerned by the organization of the resources used by a program run. What we want is to develop a spatial point of view on the entities built by the programmer when he designs his programs. From this perspective, a program must be seen as a space where computation occurs and a computation can be structured by spatial relationships. We hope to provide some evidences in the rest of this paper that the concept of space can be as fertile as mathematical logic for the development of programming languages. More specifically, we advocate that the concepts and tools developed for the algebraic construction and characterization of shapes1 provide interesting teaching, heuristic and technical alternatives to develop new data structures and new control structures for programming.

The rest of this paper is organized as follows. Section 2 and section 3 provide an informal discussion to convince the reader of the interest of introducing a topological point of view in programming. This approach is illustrated through the experimental programming language MGS used as a vehicle to investigate and validate the topological approach.

Section 2 introduces the idea of seeing a data structure as a space where the computation and the values move. Section 3 follows the spatial metaphor and presents control structures as path specifications. The previous ideas underlie MGS. Section 4 sketches this language. The presentation is restricted to the notions needed to follow the examples in the next section. Section 5 gives some examples and introduces the (DS)2 class of dynamical systems which exhibit a dynamical structure. Such kind of systems are hard to model and simulate because the state space must be computed jointly with the running state of the system. To conclude in section 6 we indicate some of the related work and we mention briefly some perspectives on the use of spatial notions.

Data Structures as Spaces 2

The relative accessibility from one element to another is a key point considered in a data structure definition:

-In a simply linked list, the elements are accessed linearly (the second after the first, the third after the second, etc.). -In a circular buffer, or in a double-linked list, the computation goes from one element to the following or to the previous one. -From a node in a tree, we can access the sons.

-The neighbors of a vertex V in a graph are visited after V when traveling through the graph. -In a record, the various fields are locally related and this localization can be named by an identifier. -Neighborhood relationships between array elements are left implicit in the array data-structure. Implementing neighborhood on arrays relies on an index algebra: index computations are used to code the access to a neighbor.

The standard example of index algebra is integer tuples with linear mappings λx.x ± 1 along each dimension (called "Von Neumann" or "Moore" neighborhoods).

This accessibility relation defines a logical neighborhood. The concept of logical neighborhood in a data structure is not only an abstraction perceived by the programmer and vanishing at the execution, but it does have an actual meaning for the computation. Very often the computation indeed complies with the logical neighborhood of the data elements and it is folk's knowledge that most of the algorithms are structured either following the structure of the input data or the structure of the output data. Let us give some examples. The recursive definition of the fold function on lists propagates an action to be performed along the traversal of a list. More generally, recursive computations on data structures respect so often the logical neighborhood, that standard highorder functions (e.g. primitive recursion) can be automatically defined from the data structure organization (think about catamorphisms and other polytypic functions on inductive types [START_REF] Meijer | Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire[END_REF][START_REF] Jeuring | Polytypic programming[END_REF]).

The list of examples can be continued to convince ourselves that a notion of logical neighborhood is fundamental in the definition of a data structure. So to define a data organization, we adopt a topological point of view: a data structure can be seen as a space, the set of positions between which the computation moves. Each position possibly holds a value3 . The set of positions is called the container and the values labeling the positions constitute the content.

This topological approach is constructive: one can define a data type by the set of moves allowed in the data structure. An example is given by the notion of "Group Based Fields" or GBF in short [START_REF] Giavitto | Group based fields[END_REF][START_REF] Giavitto | Declarative definition of group indexed data structures and approximation of their domains[END_REF]. In a uniform data structure, i.e. in a data structure where any elementary move can be used against any position, the set of moves possesses the structure of a mathematical group G. The neighborhood relationship of the container corresponds to the Cayley graph of G. In this paper, we will use only two very simple groups G corresponding to the moves |north> and |east> allowed in the usual two-dimensional grid and to the moves allowed in the hexagonal lattice figured at the right of Fig. 3.

Control Structures as Paths

In the previous section, we suggested looking at data structure as spaces in which computation moves. Then, when the computation proceeds, a path in the data structure is traversed. This path is driven by the control structures of the program. So, a control structure can be seen as a path specification in the space of a data structure. We elaborate on this idea into two directions: concurrent processes and multi-agent systems.

Homotopy of a Program Run

Consider two sequential processes A and B that share a semaphore s. The current state of the parallel execution P = A || B can be figured as a point in the plane A × B where A (resp. B) is the sequence of instructions of A (resp. B). Thus, the running of P corresponds to a path in the plane A × B. However, there are two constraints on paths that represent the execution of P. Such a path must be "increasing" because we suppose that at least one of the two subprocesses A or B must progress. The second constraint is that the two subprocesses cannot be simultaneously in the region protected by the semaphore s. This constraint has a clear geometrical interpretation: the increasing paths must avoid an "obstruction region", see Fig. 1. Such representation is known at least from the 1970's as "progress graph" [START_REF] Coffman | System deadlocks[END_REF] and is used to study the possible deadlocks of a set of concurrent processes.

Homotopy (the continuous deformation of a path) can be adapted to take into account the constraint of increasing paths and provides effective tools to detect deadlocks or to classify the behavior of a parallel program (for instance

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✁✂ ✁✂ ✁✂ ✂ ✁✂ ✁✂ ✁✂ ✂ ✁✂ ✁✂ ✁✂ ✂ ✁✂ ✁✂ ✁✂ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ✄ ✁✄ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ ☎ ✁☎ V(r) P(r) P(s) V(s) P(r) V(r) V(s) P(s) A B α β P(s) V(s) P(s) V(s) A B Fig. 1. Left:
The possible path taken by the process A || B is constrained by the obstruction resulting of a semaphore shared between the processes A and B. Right: The sharing of two semaphores between two processes may lead to deadlock (corresponding to the domain α) or to the existence of a "garden of Eden" (the domain β cannot be accessed from outside β and can only be leaved.)

in the previous example, there are two classes of paths corresponding to executions where the process A or B enters the semaphore first). Refer to [START_REF] Goubault | Geometry and concurrency: A user's guide[END_REF] for an introduction to this domain.

The Topological Structure of Interactions 4

In a multi-agent system (or an object based or an actor system), the control structures are less explicit and the emphasis is put on the local interaction between two (sometimes more) agents. In this section, we want to show that the interactions between the elements of a system exhibit a natural topology. The starting point is the decomposition of a system into subsystems defined by the requirement that the elements into the subsystems interact together and are truly independent from all other subsystems parallel evolution.

In this view, the decomposition of a system S into subsystems S 1 , S 2 , . . . , S n is functional : state s i (t + 1) of the subsystem S i depends solely of the previous state s i (t). However, the decomposition of S into the S i can depend on the time steps. So we write S t = {S t 1 , S t 2 , . . . , S t nt } for the decomposition of the system S at time t and we have:

s i (t + 1) = h t i (s i (t))
where the h t i are the "local" evolution functions of the S t i . The "global" state s(t) of the system S can be recovered from the "local" states of the subsystems: there is a function ϕ t such that s(t) = ϕ t (s 1 (t), . . . , s nt (t)) which induces a relation between the "global" evolution function h and the local evolution functions:

s(t + 1) = h(s(t)) = ϕ t (h t 1 (s 1 (t)), . . . , h t nt (s nt (t))
). The successive decomposition S t 1 , S t 2 , . . . , S t nt can be used to capture the elementary parts and the interaction structure between these elementary parts of S. Cf. Figure 2. Two subsystems S ′ and S ′′ of S interact if there are some t such that S ′ , S ′′ ∈ S t . Two subsystems S ′ and S ′′ are separable if there are some t such that S ′ ∈ S t and S ′′ ∈ S t or vice-versa. This leads to consider the set S, called the interaction structure of S, defined by the smaller set closed by intersection that contains the S t j .

Set S has a topological structure: S corresponds to an abstract simplicial complex. An abstract simplicial complex [START_REF] Henle | A combinatorial introduction to topology[END_REF] is a collection S of finite nonempty set such that if A is an element of S, so is every nonempty subset of A. The element A of S is called a simplex of S; its dimension is one less that the number of its elements. The dimension of S is the largest dimension of one of its simplices. Each nonempty subset of A is called a face and the vertex set V (S), defined by the union of the one point elements of S, corresponds to the elementary functional parts of the system S. The abstract simplicial complex notion generalizes the idea of graph: a simplex of dimension 1 is an edge that links two vertices, a simplex f of dimension 2 can be thought of as a surface whose boundaries are the simplices of dimension 1 included in f , etc.

...

S s(0)

S 1 1 s(1) S 0 1 S 1 i s(t) S ′ ∈ V (S)
Fig. 2. The interaction structure of a system S resulting from the subsystems of elements in interaction at a given time step.

MGS Principles

The two previous sections give several examples to convince the reader that a topological approach of the data and control structures of a program present some interesting perspectives for language design: a data structure can be defined as a space (and there are many ways to build spaces) and a control structure is a path specification (and there are many ways to specify a path).

Such a topological approach is at the core of the MGS project. Starting from the analysis of the interaction structure in the previous section, our idea is to define directly the set S with its topological structure and to specify the evolution function h by specifying the set S t i and the functions h t i :

the interaction structure S is defined as a new kind of data structures called topological collections; -a set of functions h t i together with the specification of the S t i for a given t are called a transformation.

We will show that this abstract approach enables an homogeneous and uniform handling of several computational models including cellular automata (CA), lattice gas automata, abstract chemistry, Lindenmayer systems, Paun systems and several other abstract reduction systems.

These ideas are validated by the development of a language also called MGS. This language embeds a complete, strict, impure, dynamically or statically typed functional language.

Topological Collections

The distinctive feature of the MGS language is its handling of entities structured by abstract topologies using transformations [START_REF] Giavitto | The topological structures of membrane computing[END_REF]. A set of entities organized by an abstract topology is called a topological collection. Here, topological means that each collection type defines a neighborhood relation inducing a notion of subcollection. A subcollection S ′ of a collection S is a subset of connected elements of S and inheriting its organization from S. Beware that by "neighborhood relation" we simply mean a relationship that specify if two elements are neighbors. From this relation, a cellular complex can be built and the classical "neighborhood structure" in terms of open and closed sets can be recovered [START_REF] Sorkin | A finitary substitute for continuous topology[END_REF].

A topological collection can be thought as a function with a finite support from a set of positions (the elements of V (S)) to a set of values (the support of a function is the set of elements on which the function takes a well defined value). Such a data structure is called a data field [START_REF] Giavitto | A framework for the recursive definition of data structures[END_REF]. This point of view is only an abstraction: the data structure is not really implemented as a function. This approach makes a distinction between the content and the container. The notions of shape [START_REF] Jay | A semantics for shape[END_REF] and shape type [START_REF] Fradet | Shape types[END_REF] also separate the set of positions of a data structure from the values it contains. Often there is no need to distinguish between the positions and their associated values. In this case, we use the term "element of the collection".

Collection Types. Different predefined and user-defined collection types are available in MGS, including sets, bags (or multisets), sequences, Cayley graphs of Abelian groups (which include several unbounded, circular and twisted grids), Delaunay triangulations, arbitrary graphs, quasi-manifolds [START_REF] Spicher | A topological framework for the specification and the simulation of discrete dynamical systems[END_REF] and some other arbitrary topologies specified by the programmer.

Building Topological Collections. For any collection type T, the corresponding empty collection is written ():T. The join of two collections C 1 and C 2 (written by a comma: C 1 ,C 2 ) is the main operation on collections. The comma operator is overloaded in MGS and can be used to build any collection (the type of the arguments disambiguates the collection built). So, the expression 1, 1+2, 2+1, ():set builds the set with the two elements 1 and 3, while the expression 1, 1+2, 2+1, ():bag computes a bag (a set that allows multiple occurrences of the same value) with the three elements 1, 3 and 3. A set or a bag is provided with the following topology: in a set or a bag, any two elements are neighbors. To spare the notations, the empty sequence can be omitted in the definition of a sequence: 1, 2, 3 is equivalent to 1, 2, 3,():seq.

Transformations

The MGS experimental programming language implements the idea of transformations of topological collections into the framework of a functional language: collections are just new kinds of values and transformations are functions acting on collections and defined by a specific syntax using rules. Transformations (like functions) are first-class values and can be passed as arguments or returned as the result of an application.

The global transformation of a topological collection s consists in the parallel application of a set of local transformations. A local transformation is specified by a rule r that specifies the replacement of a subcollection by another one. The application of a rewriting rule σ ⇒ f (σ, ...) to a collection s:

1. selects a subcollection s i of s whose elements match the pattern σ, 2. computes a new collection s ′ i as a function f of s i and its neighbors, 3. and specifies the insertion of s ′ i in place of s i into s.

One should pay attention to the fact that, due to the parallel application strategy of rules, all distinct instances s i of the subcollections matched by the σ pattern are "simultaneously replaced" by the f (s i ).

Path Pattern. A pattern σ in the left hand side of a rule specifies a subcollection where an interaction occurs. A subcollection of interacting elements can have an arbitrary shape, making it very difficult to specify. Thus, it is more convenient (and not so restrictive) to enumerate sequentially the elements of the subcollection. Such enumeration will be called a path.

A path pattern Pat is a sequence or a repetition Rep of basic filters. A basic filter BF matches one element. The following fragment of the grammar of path patterns reflects this decomposition:

Pat ::= Rep | Rep , Pat Rep ::= BF | BF /exp BF ::= cte | id | <undef>
where cte is a literal value, id ranges over the pattern variables and exp is a boolean expression. The following explanations give an interpretation for these patterns:

literal: a literal value cte matches an element with the same value. empty element the symbol <undef> matches an element whose position does not have an associated value. variable: a pattern variable a matches exactly one element with a well defined value. The variable a can then occur elsewhere in the rest of pattern or in the r.h.s. of the rule and denotes the value of the matched element. neighbor: b, p is a pattern that matches a path which begins by an element matched by b and continues by a path matched by p, the first element of p being a neighbor of b. guard: p/exp matches a path matched by p when the boolean expression exp evaluates to true.

Elements matched by basic filters in a rule are distinct. So a matched path is without self-intersection. The identifier of a pattern variable can be used only once as a basic filter. That is, the path pattern x,x is forbidden. However, this pattern can be rewritten for instance as: x,y / y = x.

Right Hand Side of a Rule. The right hand side of a rule specifies a collection that replaces the subcollection matched by the pattern in the left hand side.

There is an alternative point of view: because the pattern defines a sequence of elements, the right hand side may be an expression that evaluates to a sequence of elements. Then, the substitution is done element-wise: element i in the matched path is replaced by the element i in the r.h.s. This point of view enables a very concise writing of the rules.

A Very Simple Transformation. The map function which applies a function to each element of a collection is an example of a simple transformation:

trans map[f=\z.z] = { x => f(x)
} This transformation is made of only one rule. The syntax must be obvious (the default value of the optional parameter f is the identity written using a lambdanotation). This transformation implements a map since each element e of the collection is matched by the pattern x and will be replaced by f(e) in a parallel application strategy of the rule.

Examples

The modeling of Dynamical Systems

In this section, we show through one example the ability of MGS to concisely and easily express the state of a dynamical system and its evolution function. More examples can be found on the MGS web page and include: cellular automata-like examples (game of life, snowflake formation, lattice gas automata...), various resolutions of partial differential equations (like the diffusion-reaction à la Turing), Lindenmayer systems (e.g. the modeling of the heterocysts differentiation during Anabaena growth), the modeling of a spatially distributed signaling pathway, the flocking of birds, the modeling of a tumor growth, the growth of a meristem, the simulation of colonies of ants foraging for food, etc.

The example given below is an example of a discrete "classical dynamical system". We term it "classical" because it exhibits a static structure: the state of the system is statically described and does not change with the time. This situation is simple and arises often in elementary physics. For example, a falling stone is statically described by a position and a velocity and this set of variables does not change (even if the value of the position and the value of the velocity change in the course of time). However, in some systems, it is not only the values of state variables, but also the set of state variables and/or the evolution function, that changes over time. We call these systems dynamical systems with a dynamic structure following [START_REF] Giavitto | Mgs: a rule-based programming language for complex objects and collections[END_REF], or (DS) 2 in short. As pointed out by [START_REF] Giavitto | Modelling and Simulation of biological processes in the context of genomics, chapter[END_REF], many biological systems are of this kind. The rationale and the use of MGS in the simulation of (DS) 2 is presented in [START_REF] Giavitto | Invited talk: Topological collections, transformations and their application to the modeling and the simulation of dynamical systems[END_REF][START_REF] Giavitto | Modelling and Simulation of biological processes in the context of genomics, chapter[END_REF]. Fig. 3. From left to right: the final state of a DLA process on a torus, a chess pawn, a Klein's bottle and an hexagonal meshes. The chess pawn is homeomorphic to a sphere and the Klein's bottle does not admit a concretization in Euclidean space. These two topological collections are values of the quasi-manifold type. Such collection are build using G-map, a data-structure widely used in geometric modeling [START_REF] Lienhardt | Topological models for boundary representation : a comparison with n-dimensional generalized maps[END_REF]. The torus and the hexagonal mesh are GBFs.

Diffusion Limited Aggreation (DLA). DLA, is a fractal growth model studied by T.A. Witten and L.M. Sander, in the eighties. The principle of the model is simple: a set of particles diffuses randomly on a given spatial domain. Initially one particle, the seed, is fixed. When a mobile particle collides a fixed one, they stick together and stay fixed. For the sake of simplicity, we suppose that they stick together forever and that there is no aggregate formation between two mobile particles. This process leads to a simple CA with an asynchronous update function or a lattice gas automata with a slightly more elaborate rule set. This section shows that the MGS approach enables the specification of a simple generic transformation that can act on arbitrary complex topologies.

The transformation describing the DLA behavior is really simple. We use two symbolic values 'free and 'fixed to represent respectively a mobile and a fixed particle. There are two rules in the transformation:

1. the first rule specifies that if a diffusing particle is the neighbor of a fixed seed, then it becomes fixed (at the current position); 2. the second one specifies the random diffusion process: if a mobile particle is neighbor of an empty place (position), then it may leave its current position to occupy the empty neighbor (and its current position is made empty).

Note that the order of the rules is important because the first has priority over the second one. Thus, we have :

trans dla = { 'free, 'fixed => 'fixed, 'fixed 'free, <undef> => <undef>, 'free }
This transformation is polytypic and can be applied to any kind of collection, see Fig. 3 for a few results.

Programming in the Small: Algorithmic Examples

The previous section advocates the adequation of the MGS programming style to model and simulate various dynamical systems. However, it appears that the MGS programming style is also well fitted for the implementation of algorithmic tasks. In this section, we show some examples that support this assertion. More examples can be found on the MGS web page and include: the analysis of the Needham-Schroeder public-key protocol [START_REF] Michel | An Analysis of a Public-Key Protocol with Membranes[END_REF], the Eratosthene's sieve, the normalization of boolean formulas, the computation of various algorithms on graphs like the computation of the shortest distance between two nodes or the maximal flow, etc.

Gamma and the Chemical Computing Metaphor. In MGS, the topology of a multiset is the topology of a complete connected graph: each element is the neighbor of any other element. With this topology, transformations can be used to easily emulate a Gamma transformations [START_REF] Banatre | A parallel machine for multiset transformation and its programming style[END_REF][START_REF] Banâtre | Gamma and the chemical reaction model: Fifteen years after[END_REF]. The Gamma transformation:

M = do rp x 1 , . . . , x n if P (x 1 , . . . , x n ) by f 1 (x 1 , . . . , x n ), . . . , f m (x 1 , . . . , x n )
is simply translated into the following MGS transformation: The optional parameter iter is a system parameter that allows the programmer to choose amongst several predefined application strategies:

trans M = { x 1 , . . . , x n / P (x 1 , . . . , x n ) => f 1 (
f [iter='fixpoint](x 0 ) computes x 1 = f (x 0 ), x 2 = f (x 1 ), ..., x n = f (x n-1
) and returns x n such that

x n = x n-1 .
As a consequence, the concise and elegant programming style of Gamma is enabled in MGS: refer to the Gamma literature for numerous examples of algorithms, from knapsack to the maximal convex hull of a set of points, through the computation of prime numbers. See also the numerous applications of multiset rewriting developped in the projects Elan [START_REF]The PROTHEO project[END_REF] and Maude [START_REF]Maude home page[END_REF].

One can see MGS as "Gamma with more structure". However, one can note that the topology of a multiset is "universal" in the following sense: it embeds any other neighborhood relationship. So, it is always possible to code (at the price of explicit coding the topological relation into some value inspected at run-time) any specific topology on top of the multiset topology. We interpret the development of "structured Gamma" [START_REF] Fradet | Structured Gamma[END_REF] from this perspective. In addition, transformations are functions and functions are first citizen values in MGS. So the higher-order features of the higher-order chemical programming style (see the article by Banâtre et al. in this volume) can be easely achieved in MGS.

Two Sorting Algorithms. A kind of bubble-sort is straightforward in MGS; it is sufficient to specify the exchange of two non-ordered adjacent elements in a sequence, see Fig. 4. The corresponding transformation is defined as:

trans BubbleSort = {
x,y / x > y ⇒ y,x } The transformation BubbleSort must be iterated until a fixpoint is reached. This is not a real a bubble sort algorithm because swapping of elements happen at arbitrary places; hence an out-of-order element does not necessarily bubble to the top in the characteristic way.

Bead sort is a new sorting algorithm [START_REF] Arulanandham | Bead-sort: A natural sorting algorithm[END_REF]. The idea is to represent positive integers by a set of beads, like those used in an abacus. Beads are attached to vertical rods and appear to be suspended in the air just before sliding down (a number is read horizontally, as a row). After their falls, the rows of numbers have been rearranged such as the smaller numbers appears on top of greater numbers, see Fig. 4. The corresponding one-line MGS program is given by the transformation:

trans BeadSort = { 'empty |north> 'bead ⇒ 'bead, 'empty } This transformation is applied on the usual grid. The constant 'empty is used to give a value to an empty place and the constant 'bead is used to represent an occupied cell. The l.h.s. of the only rule of the transformation BeadSort selects the paths of length two, composed by an occupied cell at north of an empty cell. Such a path is replaced by a path computed in the r.h.s. of the rule. The r.h.s. in this example computes a path of length two with the occupied and the empty cell swapped. This transformation uses an iterated pattern x* that matches a path (a sequence of elements neighbor two by two). The keyword self refers to the collection on which the transformation is applied, that is, the entire graph. The size of a graph returns the number of its vertices. So, if the length of the path x is the same as the number of vertices in the graph, then the path x is an Hamiltonian path because matched paths are simple (no repetition of an element). The second guard prints the Hamiltonian path as a side effect and returns its argument which is not a false value. Then the third guard is checked and returns false, thus, the r.h.s. of the rule is never triggered (the ! operator introduces an assertion and !(false) raises an exception that stops the evaluation process if it is evaluated). The matching strategy ensures a maximal rule application. In other words, if a rule is not triggered, then there is no instance of a possible path that fulfills the pattern. This property implies that the previous rule must be checked on all possible Hamiltonian paths and H(g) prints all the Hamiltonian path in g before returning g unchanged.

Current Status and Related Work

The topological approach we have sketched here is part of a long term research effort [START_REF] Giavitto | Group based fields[END_REF] developed for instance in [START_REF] Giavitto | A framework for the recursive definition of data structures[END_REF] where the focus is on the substructure, or in [START_REF] Giavitto | Declarative definition of group indexed data structures and approximation of their domains[END_REF] where a general tool for uniform neighborhood definition is developed. Within this long term research project, MGS is an experimental language used to investigate the idea of associating computations to paths through rules. The application of such rules can be seen as a kind of rewriting process on a collection of objects organized by a topological relationship (the neighborhood). A privileged application domain for MGS is the modeling and simulation of dynamical systems that exhibit a dynamic structure.

Multiset transformation is reminiscent of multiset-rewriting (or rewriting of terms modulo AC). This is the main computational device of Gamma [START_REF] Banatre | A parallel machine for multiset transformation and its programming style[END_REF], a language based on a chemical metaphor; the data are considered as a multiset M of molecules and the computation is a succession of chemical reactions according to a particular rule. The CHemical Abstract Machine (CHAM) extends these ideas with a focus on the expression of semantic of non deterministic processes [START_REF] Berry | The chemical abstract machine[END_REF]. The CHAM introduces a mechanism to isolate some parts of the chemical solution. This idea has been seriously taken into account in the notion of P systems. P systems [START_REF] Paun | From cells to computers: Computing with membranes (P systems)[END_REF] are a recent distributed parallel computing model based on the notion of a membrane structure. A membrane structure is a nesting of cells represented, e.g, by a Venn diagram without intersection and with a unique superset: the skin. Objects are placed in the regions defined by the membranes and evolve following various transformations: an object can evolve into another object, can pass trough a membrane or dissolve its enclosing membrane. As for Gamma, the computation is finished when no object can further evolve. By using nested multisets, MGS is able to emulate more or less the notion of P systems. In addition, patterns like the iteration + go beyond what is possible to specify in the l.h.s. of a Gamma rule.

Lindenmayer systems [START_REF] Lindenmayer | Mathematical models for cellular interaction in development, Parts I and II[END_REF] have long been used in the modeling of (DS) 2 (especially in the modeling of plant growing). They loosely correspond to transformations on sequences or string rewriting (they also correspond to tree rewriting, because some standard features make particularly simple to code arbitrary trees, Cf. the work of P. Prusinkiewicz [START_REF] Prusinkiewicz | L systems: from formalism to programming languages[END_REF]). Obviously, L systems are dedicated to the handling of linear and tree-like structures.

There are strong links between GBF and cellular automata (CA), especially considering the work of Z. Róka which has studied CA on Cayley graphs [START_REF] Róka | One-way cellular automata on Cayley graphs[END_REF]. However, our own work focuses on the construction of Cayley graphs as the shape of a data structure and we develop an operator algebra and rewriting notions on this new data type. This is not in the line of Z. Róka which focuses on synchronization problems and establishes complexity results in the framework of CA.

A unifying theoretical framework can be developed [START_REF] Giavitto | MGS: a programming language for the transformations of topological collections[END_REF][START_REF] Giavitto | The topological structures of membrane computing[END_REF], based on the notion of chain complex developed in algebraic combinatorial topology. However, we do not claim that we have achieved a useful theoretical framework encompassing the previous paradigms. We advocate that few topological notions and a single syntax can be consistently used to allow the merging of these formalisms for programming purposes.

The current MGS interpreter is freely available at the MGS home page: mgs.lami.univ-evry.fr. A compiler is under development where a static type discipline can be enforced [START_REF] Cohen | Typing rule-based transformations over topological collections[END_REF][START_REF] Cohen | Typage fort et typage souple des collections topologiques et des transformations[END_REF]). There are two versions of the type inference systems for MGS: the first one is a classical extension of the Hindley-Milner type inference system that handles homogeneous collections. The second one is a soft type system able to handle heterogeneous collection (e.g. a sequence containing both integers and booleans is heterogeneous).

  x 1 , . . . , x n ), . . . , f m (x 1 , . . . , x n ) } and the application M(b) of a Gamma transformation M to a multiset b is replaced in MGS by the computation of the fixpoint iteration M[iter='fixpoint](b).

Fig. 4 .

 4 Fig. 4. Left: Bubble sort. Right: Bead sort [1].

G. Gaston-Granger in[START_REF] Granger | La pense de l'espace[END_REF] considers three avenues in the formalization of the concept of space: shape (the algebraic construction and the transformation of space and spatial configurations), texture (the continuum) and measure (the process of counting and coordinatization[START_REF] Weyl | The Classical Groups (their invariants and representations)[END_REF]). In this work, we rely on elementary concepts developed in the field of combinatorial algebraic topology for the construction of spaces[START_REF] Henle | A combinatorial introduction to topology[END_REF].

The ideas exposed in this section are developed in[START_REF] Giavitto | Data structure as topological spaces[END_REF][START_REF] Giavitto | Invited talk: Topological collections, transformations and their application to the modeling and the simulation of dynamical systems[END_REF].

A point in space is a placeholder awaiting for an argument, L. Wittgenstein, (Tractatus Logico Philosophicus, 2.0131).

This section is adapted from[START_REF] Spicher | A topological framework for the specification and the simulation of discrete dynamical systems[END_REF].

Acknowledgments

The authors would like to thanks Franck Delaplace at LaMI, Frédéric Gruau at University of Paris-Sud, Florent Jacquemard at INRIA/LSV-Cachan, C. Godin and P. Barbier de Reuille at CIRAD-Montpellier, Pierre-Etienne Moreau at Loria-Nancy, Éric Goubault at CEA-Saclay, P. Prusinkiewicz at the University of Calgary (who coined the term "computation in space") and the members of the Epigenomic group at GENOPOLE-Évry, for stimulating discussions, thoughtful remarks and warm support. We gratefully acknowledge the financial support of the CNRS, the GDR ALP, IMPBIO, the University of Évry and GENOPOLE.


Mathieu Rospabé
email: mrospabe@jamstec.go.jp

Georges Ceuleneer

Mathieu Benoit

Mary-Alix Kaczmarek

inclusions in chromites from the dunitic mantlecrust transition (Oman ophiolite) reveal high temperature fluid-melt-rock interaction controlled by faulting

Keywords: silicate inclusions, disseminated chromites, variably impregnated dunites, dunitic mantle-crust transition zone, fluid-melt-peridotite interaction, hybridization, Maqsad, Oman ophiolite

Composition gradients in silicate inclusions in chromites

from the dunitic mantle-crust transition (Oman ophiolite) reveal high temperature fluid-melt-rock interaction controlled by faulting

INTRODUCTION

The boundary between the mantle and the oceanic crust is known to be a reactive transition where significant interactions occur between the melts issued from the deep mantle partial melting, whatever their composition, and the peridotites from the uppermost mantle. These processes, called melt-rock reactions, lead to mineral resorption and/or precipitation (i.e. modal metasomatism) as well as to chemical re-equilibration between the migrating melts and the surrounding rocks (i.e. cryptic metasomatism) (e.g. [START_REF] Kelemen | Reaction between ultramafic rock and fractionating basaltic magma I. phase relations, the origin of calc-alkaline magma series, and the formation of discordant dunite[END_REF][START_REF] Kelemen | Extraction of mid-ocean-ridge basalt from the upwelling mantle by focused flow of melt in dunite channels[END_REF][START_REF] Constantin | Gabbroic intrusions and magmatic metasomatism in harzburgites from the Garrett transform fault: implications for the nature of the mantle-crust transition at fast-spreading ridges[END_REF][START_REF] Godard | Relationships between geochemistry and structure beneath a palaeo-spreading centre: A study of the mantle section in the Oman ophiolite[END_REF][START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF][START_REF] Sanfilippo | Mantle-crust interactions in the oceanic lithosphere: Constraints from minor and trace elements in olivine[END_REF]2016;[START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF], and thus may have a big impact on the chemical and material transfers below oceanic spreading centres in different settings and at various expansion rates. Dunites and chromitites are frequently observed at the mantle-crust transition both in ophiolites and in present-day oceans, and look to be two important petrological products of melt-peridotite reaction. In this context, dunite is mainly interpreted as a former mantle harzburgite modified by interaction with a melt migrating interstitially along grain boundaries: due to the undersaturated-in-silica character of the melt at low pressure, it induces the dissolution of pyroxenes and the concomitant precipitation of olivine, leading to a 'residual', actually reactive, product made of olivine and minor chrome-spinel [START_REF] Dick | Evidence of partial melting in the Josephine peridotite[END_REF][START_REF] Quick | The origin and significance of large, tabular dunite bodies in the Trinity peridotite, northern California[END_REF][START_REF] Kelemen | Reaction between ultramafic rock and fractionating basaltic magma I. phase relations, the origin of calc-alkaline magma series, and the formation of discordant dunite[END_REF]1992).

One hypothesis calls for a hydrated environment for this dunitization process below oceanic Journal Pre-Proof spreading centres, a melt hybrid between the MORB and a hydrated component enhancing the transformation of the former mantle harzburgite and the melting out of residual orthopyroxene [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF][START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]. Recent studies in the Oman ophiolite highlighted the importance of the development of deep-seated synmagmatic faults i) for the accretion of the oceanic lithosphere, triggering the deep introduction of seawater that interacted with the magmatic system [START_REF] Abily | Synmagmatic normal faulting in the lower oceanic crust: Evidence from the Oman ophiolite[END_REF], and ii) on the genesis of such hybrid melts along these avenues for fluids (Rospabé et al., 2019a;2019b). On the other hand, the presence of hydrous silicate inclusions in chromite grains from Oman and elsewhere is a long-standing observation likewise calling for the hydrated formation environment of chromitites (e.g. [START_REF] Dunlop | Origin of chromite deposits in ophiolitic complexes: evidence for a volatile-and sodium-rich reducing fluid phase[END_REF]2017;[START_REF] Talkington | Platinum-group minerals and other solid inclusions in chromite of ophiolitic complexes: Occurrence and petrological significance[END_REF][START_REF] Matveev | Role of water in the origin of podiform chromitite deposits[END_REF][START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF]. Therefore, it suggests common fluid-melt hybridization processes, or fluid-melt-peridotite reactions, may account for the formation of both dunites and chromitite beneath oceanic spreading centres.

In the Maqsad area of the Oman ophiolite, the lithosphere accreted in a MORB magmatic environment [START_REF] Ceuleneer | Thermal structure of a fossil mantle diapir inferred from the distribution of mafic cumulates[END_REF][START_REF] Benoit | Trace element and isotopic characterization of mafic cumulates in a fossil mantle diapir (Oman ophiolite)[END_REF][START_REF] Kelemen | Geochemistry of gabbro sills in the crust-mantle transition zone of the Oman ophiolite: implications for the origin of the oceanic lower crust[END_REF][START_REF] Korenaga | Origin of gabbro sills in the Moho transition zone of the Oman ophiolite: Implications for magma transport in the oceanic lower crust[END_REF][START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF][START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF] and the hydrated features look restricted to the dunitic mantle-crust transition zone (DTZ) [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF][START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]. In this area, the nature and chemical compositions of magmatic impregnations in dunites (i.e. crystallized from interstitial melts percolating between olivine grains) and of silicate inclusions in chromite grains, disseminated at the scale of the whole DTZ or concentrated in chromitite ore bodies, are very akin [START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF][START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF]. However, no study was fully dedicated to the detailed study of such silicate inclusions in chromite grains in a well-established petrological, geochemical and structural frame, while they may provide essential information about the fluid-meltperidotite reaction at the mantle-crust transition and thus on the relationship between dunite and chromitite.

Journal Pre-Proof

In this article, we present a detailed description of the silicate inclusions in chromite along a near 400 m-thick cross-section we already characterized for its petrology, geochemistry and structural features (Rospabé et al., 2019a). The main purposes of this study are: i) to explore the similarities and/or distinctions between the silicate inclusions in chromite grains disseminated in the DTZ on one hand, and the mineral impregnations partly forming the matrix of the host dunites on the other hand; and ii) to test the hypothesis of the possible influence of the synmagmatic faults on the hybridization of the fluids/melts that have dissolved and fractionated the Cr at the mantle-oceanic crust transition.

GEOLOGICAL BACKGROUND AND SAMPLES

The Oman ophiolite is the largest oceanic lithosphere fragment preserved from the Tethys ocean. It accreted along a possibly fast spreading centre circa 95-97 million years ago contemporaneously, soon before or soon after the initiation of the intra-oceanic thrusting event [START_REF] Boudier | Shear zones, thrusts and related magmatism in the Oman ophiolite: Initiation of thrusting on an oceanic ridge[END_REF][START_REF] Montigny | K-Ar and Ar study of metamorphic rocks associated with the Oman ophiolite: Tectonic implications[END_REF][START_REF] Warren | Dating the geologic history of Oman's Semail ophiolite: Insights from U-Pb geochronology[END_REF][START_REF] Rioux | Tectonic development of the Samail ophiolite: High-precision U-Pb zircon geochronology and Sm-Nd isotopic constraints on crustal growth and emplacement[END_REF]2016) that preceded its final obduction onto the Arabian margin ∼ 70 Ma ago [START_REF] Glennie | Late Cretaceous nappes in Oman Mountains and their geologic evolution[END_REF]. At the scale of the ophiolite, the spatial distribution of the petrological nature of i) mafic dikes cross cutting the harzburgitic mantle section, and ii) cumulates from the lower crustal section, attests to the coexistence of both MORB and depleted calc-alkaline series at the time of the igneous accretion of the ophiolite [START_REF] Benoit | The remelting of hydrothermally altered peridotite at mid-ocean ridges by intruding mantle diapirs[END_REF][START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF][START_REF] Python | Chromian spinels in mafic-ultramafic mantle dykes: Evidence for a two-stage melt production during the evolution of the Oman ophiolite[END_REF][START_REF] Clénet | Thick sections of layered ultramafic cumulates in the Oman ophiolite revealed by an airborne hyperspectral survey: Petrogenesis and relationship to mantle diapirism[END_REF][START_REF] Abily | Caractéristiques pétrographique, géochimique et structurale de la section crustale profonde de l'ophiolite d'Oman : implications pour la genèse des magmas et le fonctionnement des chambres magmatiques à l'aplomb d'un centre d'expansion océanique[END_REF].

The largest MORB segment is located in the SE of the ophiolite [START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF][START_REF] Python | Chromian spinels in mafic-ultramafic mantle dykes: Evidence for a two-stage melt production during the evolution of the Oman ophiolite[END_REF]. It was centred on and fed with melts by a mantle uprising structure now frozen in the Sumail massif (evidenced by the high-temperature plastic deformation pattern recorded in harzburgites) and known as the Maqsad diapir [START_REF] Rabinowicz | Melt segregation and flow in mantle diapirs below spreading centers: evidence from the Oman ophiolite[END_REF][START_REF] Ceuleneer | Mantle flow patterns at an oceanic spreading centre: The Oman peridotites record[END_REF][START_REF] Ceuleneer | Evidence for a Paleo-Spreading Center in the Oman Ophiolite: Mantle Structures in the Maqsad Area[END_REF][START_REF] Jousselin | Detailed mapping of a mantle diapir below a paleospreading center in the Oman ophiolite[END_REF]. The strong igneous activity related to the rise of the Maqsad diapir is attested by: i) abundant melt migration structures cropping out within the residual, harzburgitic mantle section [START_REF] Ceuleneer | Thermal structure of a fossil mantle diapir inferred from the distribution of mafic cumulates[END_REF][START_REF] Benoit | Trace element and isotopic characterization of mafic cumulates in a fossil mantle diapir (Oman ophiolite)[END_REF][START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF]; Journal Pre-Proof ii) the thick dunitic mantle-crust transition zone (DTZ) especially atop the diapiric structure (> 300 m) [START_REF] Ceuleneer | Structures in podiform chromite from the Maqsad district (Sumail ophiolite, Oman)[END_REF][START_REF] Boudier | Nature of the moho transition zone in the Oman ophiolite[END_REF][START_REF] Jousselin | Detailed mapping of a mantle diapir below a paleospreading center in the Oman ophiolite[END_REF][START_REF] Abily | The dunitic mantle-crust transition zone in the Oman ophiolite: Residue of melt-rock interaction, cumulates from high-mgo melts, or both?[END_REF][START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]; and iii) numerous chromitite ore bodies observed in both the DTZ and the mantle section (in this case systematically surrounded by a dunitic aureole) [START_REF] Ceuleneer | Structures in podiform chromite from the Maqsad district (Sumail ophiolite, Oman)[END_REF].

The dunites from the Maqsad DTZ are mainly interpreted as a reactional product of melt-peridotite interactions in the uppermost harzburgitic mantle (i.e. orthopyroxene resorption) (e.g. [START_REF] Boudier | Nature of the moho transition zone in the Oman ophiolite[END_REF][START_REF] Godard | Relationships between geochemistry and structure beneath a palaeo-spreading centre: A study of the mantle section in the Oman ophiolite[END_REF][START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF][START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF]. However, a double origin cannot be ruled out since the uppermost few tens of meters of the DTZ (i.e. approaching the base of the lower gabbroic crust) have a composition consistent with a cumulative origin after olivine fractionation from a Mg-rich melt [START_REF] Abily | The dunitic mantle-crust transition zone in the Oman ophiolite: Residue of melt-rock interaction, cumulates from high-mgo melts, or both?[END_REF]. In addition to olivine, the dunites contain a large variety of interstitial minerals considered as impregnation, i.e. that crystallized from an interstitial melt [START_REF] Rabinowicz | Melt segregation and flow in mantle diapirs below spreading centers: evidence from the Oman ophiolite[END_REF][START_REF] Nicolas | Mantle-crust transition zone and origin of wehrlitic magmas: Evidence from the Oman ophiolite[END_REF][START_REF] Boudier | Nature of the moho transition zone in the Oman ophiolite[END_REF][START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF][START_REF] Abily | The dunitic mantle-crust transition zone in the Oman ophiolite: Residue of melt-rock interaction, cumulates from high-mgo melts, or both?[END_REF][START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF][START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]2019a).

Clinopyroxene and plagioclase are the most abundant impregnation minerals which is consistent with the MORB igneous environment characterizing the Maqsad area. However, the discovery of interstitial orthopyroxene and amphibole in dunites in the upper half of the DTZ mainly, while virtually absent from the mantle and crustal cumulates in this area, contributed to the development of the hypothesis of a hydrated origin of the DTZ dunites, involving fluid(s) and/or melt(s) richer in silica than the MORB [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF]. In this frame, the mantle-crust transition zone may be the place for hybridization between the MORB issued from the deep partial melting within the Maqsad diapir and a downgoing hydrated component -probably hydrothermal seawater in origin. In addition to the peculiar impregnating minerals, another witness of the hydrated magmatism environment in which both dunites and chromitites formed is the frequent presence of hydrous silicate Journal Pre-Proof minerals in inclusion in chromite (amphibole, mica among others) observed both in chromitite ore bodies and in chromite grain scattered in dunites [START_REF] Lorand | Silicate and base-metal sulfide inclusions in chromites from the Maqsad area (Oman ophiolite, Gulf of Oman): A model for entrapment[END_REF][START_REF] Leblanc | Chromite crystallization in a multicellular magma flow: Evidence from a chromitite dike in the Oman ophiolite[END_REF][START_REF] Schiano | Primitive basaltic melts included in podiform chromites from the Oman Ophiolite[END_REF][START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF][START_REF] Rollinson | Polymineralic inclusions in mantle chromitites from the Oman ophiolite indicate a highly magnesian parental melt[END_REF][START_REF] Zagrtdenov | Anatomy of a chromitite dyke in the mantle/crust transition zone of the Oman ophiolite[END_REF][START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF]2019b). Furthermore, the correlation between the vertical chemical patterns across the DTZ dunites and chromitites (whole rocks, matrix minerals) and the presence of fault zones and/or magmatic breccias evidenced that the mantle-crust transition has been tectonically affected at an early magmatic stage, with limited offsets, and that such faults may have favoured the hybridization process (Rospabé et al., 2019a;2019b; see also [START_REF] Abily | Synmagmatic normal faulting in the lower oceanic crust: Evidence from the Oman ophiolite[END_REF]. Other recent studies likewise evidenced that chromitites are regularly located along or near faults or shear zones in the Oman ophiolite [START_REF] Boudier | Structural control on chromitite deposits in ophiolites: The oman case[END_REF][START_REF] Zagrtdenov | Anatomy of a chromitite dyke in the mantle/crust transition zone of the Oman ophiolite[END_REF]. All these observations allow pointing out the importance of the structural control on the igneous processes involved in the formation of both dunite and chromitite at the mantle-crust transition beneath oceanic spreading centres, at least in the context of Oman.

However, the silicate inclusions in the dispersed chromite grains, ubiquitous in the DTZ dunites, are regularly a left-over of the fluid-melt-rock reaction products, petrologists more commonly studying the dunite matrix or the most spectacular chromitite ore bodies.

In order to better explore the similarities and/or differences in the nature of the silicate inclusions in the scattered chromite grains and the impregnations in the surrounding matrix, and to investigate the role of the early synmagmatic faults on the nature and chemistry of such inclusions, we performed a detailed and systematic determination of the nature and composition of inclusions in chromite in 42 variably impregnated dunites. The samples were collected along a full cross-section that has already been described in terms of major and trace mineral and whole rock compositions as well as for its structural features (cross-section #3 in Rospabé et al., 2019a). The sampling sites extent from 685 to 1072 m in altitude, covering a large portion of the DTZ in this area. It is worth noting that, as the Journal Pre-Proof central part of (and above) the Maqsad diapir has escaped from off-axis transposition and that the Sumail massif is tilted by less than 10°, we consider the current altitudes as close to paleo-depths.

The cross-section shows a clear petrological dichotomy that is quite characteristic of the Maqsad DTZ [START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]: i) the half lower part is primarily made of pure dunites (i.e. only olivine and chromite) and of very minor cpx-and amph-bearing dunites, and locally hosts one chromitite (16OM38B, 748 m) and schlierens of chromite (16OM41B and 41C, 772 m); ii) the half upper part, basically above the altitude 850 m, is extensively impregnated with clinopyroxene, plagioclase, amphibole and orthopyroxene, allowing to describe samples as cpx-bearing-, pl-bearing-, cpx/plbearing-, opx/pl/cpx-bearing-and amph-bearing (± opx/pl/cpx) dunites [START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF] (Fig. 1a). Three main regional faults and four other minor fractures cross-cut the cross-section, sometimes corresponding to a change in the lithology from pure to impregnated dunites. These structural features were interpreted as synmagmatic faults that later evolved into hydrothermal faults (higher serpentinization degree along fault and fracture planes). They developed early before the end of the igneous activity in this area and influenced both the petrological and geochemical variability of the DTZ (Rospabé et al., 2019a). The faults present two main orientations: N130°E, parallel to the strike of the sheeted dike complex in the Sumail massif [START_REF] Pallister | Structure of the sheeted dike complex of the Samail Ophiolite near Ibra, Oman[END_REF][START_REF] Macleod | Ridge axial segmentation in the Oman ophiolite: Evidence from along-strike variations in the sheeted dyke complex[END_REF], i.e. parallel to the paleo-ridge axis and to the MORB segment fed by the Maqsad diapir [START_REF] Ceuleneer | Evidence for a Paleo-Spreading Center in the Oman Ophiolite: Mantle Structures in the Maqsad Area[END_REF][START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF] -and N165°E to N180°E, parallel to the high temperature Muqbariah shear zone limiting to the SW the area fed by mantle flux diverging from the diapir [START_REF] Amri | Tectonic setting for the genesis of oceanic plagiogranites: evidence from a paleo-spreading structure in the Oman ophiolite[END_REF].

DISTRIBUTION OF SILICATE INCLUSIONS ACROSS THE DTZ

The nature of the silicate inclusions in chromite has been determined according to their microprobe analysis. We identified amphibole (amph), clinopyroxene (cpx), diopside of different composition Journal Pre-Proof than cpx (diop; see [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF] and the description of chemistry below), mica, orthopyroxene (opx) and olivine (ol) as the major mineral phases in inclusions. They have all been observed both as monomineralic inclusions or as parts of polymineralic inclusions indifferently. Amphibole and clinopyroxene are by far the most abundant minerals in inclusion in the disseminated chromites in samples from this cross-section (Fig. 1b), representing more than 80 % of all the analyses. As minor contents, we also identified anorthitic plagioclase feldspar (pl), garnet (gt), as well as a more albitic plagioclase (alb) and one chlorapatite (Cl-ap).

We observe the following distribution across the section (Fig. 1):

• The very base -below the lower main fault (< ~ 715 m) -is one of the levels displaying the lesser abundance of inclusions along the cross-section (less than 10 inclusions per thin section) (Fig. 1c).

We can observe a little abundance of monomineralic inclusions (at least apparently in thin section) in the chromites dispersed in pure dunites, made of clinopyroxene, by far the most abundant phase (76 %), amphibole (22 %) and diopside (2 %) (Figs. 1b and2a-b). Polymineralic inclusions are globally absent apart from one assemblage clinopyroxene-mica.

• The pure dunitic interval in between the lower and the intermediate main faults (~ 715-850 m) also contains mica and olivine in inclusion in addition to amph, cpx and diop, especially near the chromiterich level around 750 to 775 m in altitude (chromitite, schlierens) (Figs. 1a-b and2c). This latter interval is also characterized by an increase of the amount of inclusions, both monomineralic and polymineralic. This increase is progressive from levels poor in inclusions to this richer level (Fig. 1c). Monomineralic inclusions are mainly clinopyroxene (57%) then amphibole (34%) (others 9%).

Polymineralic inclusions are mainly represented by the association amphibole-clinopyroxene and to a lesser extent by amphibole-diopside and amphibole-mica assemblages.

• The lower impregnated portion, in between the intermediate and the higher main faults (~ 850-990 Journal Pre-Proof m), shows the broader variety in included minerals (Fig. 1b). Mica occurs from 867 to 906 m, while this interval is also characterized by the absence of clinopyroxene in inclusion (gap between 854 and 906 m). In addition to amph, cpx, diop, mica and ol, the presence of opx (Fig. 2d-e), gt, pl as well as of one albitic plagioclase and one chlorapatite is attested, especially within the altitude range ~ 900-975 m. The albitic plagioclase was observed in association with anorthitic plagioclase within a polymineralic inclusion (Fig. 2f). The Cl-apatite is associated to orthopyroxene, amphibole partly altered into tremolite, serpentine (derived from opx alteration) and Fe-Ni sulphides (Fig. 2g). In these opx/pl/cpx-bearing-and amph-bearing (± opx/pl/cpx) dunites from the lower impregnated portion, the amount of monomineralic inclusions in chromite grains show alternating decreasing and increasing trends vertically, with a characteristic thickness of about 50 m (Fig. 1c). The higher amounts basically correlate to the intermediate main fault (~ 850 m) and to another minor fracture (~ 955 m). Along this portion, monomineralic inclusions are mainly amphibole, by far (49 %), then clinopyroxene (12 %) and orthopyroxene (6 %) (others 33%). Numerous garnet and plagioclase inclusions were analysed but almost all of them occur in a single sample (16OM16A, 918 m), that contains also the Cl-apatite and a significant proportion of orthopyroxene inclusions. Polymineralic inclusions are mainly made of amphibole-mica and amphibole-clinopyroxene and of a few amphibole-orthopyroxene assemblages along the lower impregnated portion (Fig. 2h).

• The upper impregnated portion, above the higher main fault (> ~ 990 m), corresponds to the altitude level hosting a neighbouring stratiform chromitite ore body [START_REF] Ceuleneer | Structures in podiform chromite from the Maqsad district (Sumail ophiolite, Oman)[END_REF][START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF][START_REF] Rollinson | Mantle podiform chromitites do not form beneath mid-ocean ridges: A case study from the Moho transition zone of the Oman ophiolite[END_REF]Rospabé et al., 2019b). The inclusion content decreases upsection along this upper part of the section (Fig. 1c). Monomineralic inclusions are mainly clinopyroxene (54 %) and amphibole (38 %), together with a few orthopyroxene (4 %) (others 5 %), while polymineralic inclusions are by decreasing order of occurrence amphibole-orthopyroxene, amphibole-clinopyroxene and amphibole-mica assemblages.

Journal Pre-Proof At the scale of the entire cross-section, beyond the alternating increases/decreases in the abundance of inclusions, we observe a global decrease in the amount of inclusions (especially the monomineralic ones) from the pure dunitic lower half of the cross-section, especially from the chromite-rich interval, to its impregnated top (i.e. 'Christmas tree-shape' pattern of the vertical distribution of inclusions) (Fig. 1c).

MINERAL CHEMISTRY

Analytical note

The major and minor element content of silicate inclusions and of their host chromite was measured in situ by electron probe microanalysis (EPMA), using a Cameca SX 100 (Microsonde Ouest, Brest, France) (accelerating voltage: 15 kV; beam current: 20 nA; electron beam diameter: 1 μm for all analysis; analysis counting time for each type of minerals: 10 s on peak for each element, 5 s on backgrounds on both sides of the peak). Compositions, averaged per sample for each mineral phase in inclusion, are reported in Supplementary Tables 1S-6S.

Host chromite

Dispersed chromites in dunites display variable compositions with XCr (100 × Cr/(Cr + Al)) = 45.6-63.1 mol%, XMg (100 × Mg/(Mg + Fe 2+ )) = 32.8-63.7 mol%, YFe 3+ (100 × Fe 3+ /(Cr + Al + Fe 3+ )) = 4.50-16.4 mol%, and TiO2 = 0.31-1.37 wt% (Rospabé et al., 2019a). The studied chromitite (16OM38B) has chromite with XCr = 58.4 mol%, XMg = 65.7 mol%, YFe 3+ = 5.77 mol% and TiO2 = 0.46 wt%.

Amphibole inclusions (Table 1S)

Amphibole inclusions in the dispersed chromites are pargasites to pargasitic hornblendes (~ 2/3) and magnesio-hastingsites to magnesio-hastingsitic hornblendes (~ 1/3) with no preferential distribution in regard to the host dunite type. They are characterized by XMg = 86.1-92.7 mol%, Al2O3 = 10.2-Journal Pre-Proof 14.6 wt%, Cr2O3 = 1.76-3.32 wt% and TiO2 = 0.59-4.71 wt%. The composition of amphibole spans the same variation ranges whatever it occurs as monomineralic inclusions or as parts of polymineralic inclusions (Fig. 3a) and does not show systematic variations relative to the modal content of chromite in the host dunites (Fig. 3b). The Na2O and K2O contents vary from 2.85 to 4.07 wt% and from detection limits to 0.93 wt% respectively, defining a XNa (100 × Na/(Na + K)) comprised between 86.2 and 99.8 mol%. The P2O5, Cl and F contents were also determined and show variations from detection limits to 0.12 wt%, 0.12 wt%, and 0.17 wt% in average respectively (maximum values obtained for individual analyses: P2O5 = 0.30 wt%, Cl = 0.24 wt%, F = 0.40 wt%). The analysed chromitite contains pargasitic hornblende inclusions with XMg = 93.8 mol%, Al2O3 = 11.1 wt%, Cr2O3 = 3.08 wt%, TiO2 = 1.50 wt% and XNa = 99.2 mol% (Na2O = 4.56 wt%).

Clinopyroxene -diopside inclusions (Table 2S)

According to our previous classification [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF], the distinction was made between the composition of 'clinopyroxene', akin from the one of poikilitic magmatic impregnations studied in previous works (e.g. [START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF], and the composition of 'diopside' which is intermediate between the visibly igneous cpx and hydrothermal diopside (e.g. in diopsidites from the mantle section; Python et al., 2007a;2007b;[START_REF] Abily | Caractéristiques pétrographique, géochimique et structurale de la section crustale profonde de l'ophiolite d'Oman : implications pour la genèse des magmas et le fonctionnement des chambres magmatiques à l'aplomb d'un centre d'expansion océanique[END_REF].

Clinopyroxene in inclusion in dispersed chromites is characterized by XMg = 88.3-93.8 mol%, Al2O3 = 1.60-3.08 wt%, Cr2O3 = 0.93-1.96 wt%, Na2O = 0.24-0.93 wt% and TiO2 = 0.15-0.62 wt%, with compositions similar in monomineralic vs. polymineralic inclusions and independent from the chromite content in host dunites (Fig. 3c-d). The only one clinopyroxene inclusion analysed in the chromitite has a higher XMg (94.3 mol%) and displays Al2O3 = 2.16 wt%, Cr2O3 = 1.50 wt%, Na2O = 1.15 wt% and TiO2 = 0.29 wt%.

The second type of clinopyroxene called 'diopside' was identified in a lesser extent in a few Journal Pre-Proof disseminated chromites and is characterized by averaged higher XMg (95.2 mol%) and lower Al2O3 (0.70 wt%), Cr2O3 (0.78 wt%), Na2O (0.20 wt%) and TiO2 (0.13 wt%).

Mica inclusions (Table 3S)

Mica in inclusion in dispersed chromites shows averaged compositions evolving from aspidolite (Naphlogopite) to intermediate between aspidolite and phlogopite, with a XNa comprised between 37.6 and 98.3 mol% (Na2O = 2.51-6.79 wt%; K2O = 0.17-5.45 wt%; Na2O + K2O = 4.69-7.97 wt%), but individual analyses of both monomineralic and polymineralic inclusions span the entire range between the two end-members (Fig. 3e). They are also characterized by a XMg = 90.1-94.5 wt% and by a Ti-and Cr-rich character (TiO2 = 0.73-5.06 wt%; Cr2O3 = 1.60-3.75 wt%). Mica inclusions in the chromitite are characterized by higher XMg (96.7 mol%) and lower TiO2 (1.46 wt%) averaged values. According to individual analyses, aspidolite is distinguishable from phlogopite in terms of Na2O (5.33-6.52 wt% vs. 0.35-1.90 wt%) and K2O (0.85-0.94 wt% vs. 6.14-8.81 wt%) contents, responsible of two populations in XNa in this chromitite (5.6-32.0 vs. 89.6-92.1 mol%, mean 39.3 mol%).

Orthopyroxene inclusions (Table 4S)

Orthopyroxene in inclusion in dispersed chromites is enstatite with XMg = 85.7-93.5 mol%, Al2O3 = 0.61-2.76 wt%, Cr2O3 = 0.47-1.54 and TiO2 = 0.02-0.75 wt%. As amphibole and clinopyroxene, orthopyroxene inclusions display similar compositions whatever the mode of occurrence (monomineralic and polymineralic).

Olivine inclusions (Table 5S)

Olivine in inclusion in dispersed chromites is forsterite, characterized by Fo (100 × Mg/(Mg + Fetotal)) = 88.7-94.8 mol%, NiO = 0.25-0.44 wt% and CaO = 0.02-0.16 wt%. Olivine in inclusion in the chromitite has a high Fo (94.7 mol%) and NiO (0.53 wt%) contents.

Journal Pre-Proof

Other minor inclusions (Table 6S)

Plagioclase in inclusion is anorthite, with An (100 × Ca/(Ca + Na + K)) = 86.7-90.3 mol%. The albitic, Na-rich plagioclase is characterized by an An content of 33.7 mol%.

Garnet in inclusion is grossular to hydrogrossular with a SiO2 content (22.2-39.0 wt%) strictly decreasing with the sum of oxides (R 2 = 0.9806) and CaO = 31.3-40.9 wt%, Al2O3 = 15.0-24.2 wt% and FeO = 0.58-10.0 wt%.

The Cl-rich apatite is characterized by CaO = 53.3 wt%, P2O5 = 41.9 wt%, Cl = 5.39 wt% and F = 0.68 wt%.

Chemical evolutions along the cross-section

The major element composition of chromite and of silicate inclusions shows pronounced variations along the studied cross-section, and defines distinct trends (Fig. 4). Some chemical contents/indexes vary according to the nature of host rocks, especially concerning the composition of chromite and the TiO2 content in inclusions. For instance, in pure dunites, the TiO2 content remains mainly below 0.5 wt% in chromite and below 1.5 wt% in amphibole inclusions, while in impregnated dunites it reaches values up to 1.25 wt% in chromite and 4.75 wt% in amphibole (Fig. 4a-b).

The composition of both chromite and silicate inclusions strongly correlates to each other. These variations are not randomly distributed relative to the main regional faults that affect the crosssection, as was already noted for the composition of whole rocks and minerals from the matrix and suspected for the inclusions (Rospabé et al., 2019a). Compositions are evolving continuously on tens of meters toward fault zones rather than showing sudden shifts in their immediate vicinity:

• In the pure dunitic lower part, the increase of XMg in chromite is observed approaching the main Journal Pre-Proof faults while the XCr in chromite, XMg and TiO2 and F contents in amphibole, and Na2O content in clinopyroxene decrease concomitantly (Fig. 4a-c).

• In the impregnated upper part, the approach of the main faults is marked by an increase of the Cl content in amphibole inclusions associated to the decrease of XNa in amphibole and in a lesser extent mica (i.e. increase of K2O content relative to the Na2O content) and of the TiO2 content in chromite and all silicate inclusions (Fig. 4a-f). These variations are strongest in the vicinity of the upper main fault with a clear common increase of the P2O5 and the F contents in amphibole, together with the strongest increase in the Cl content (Fig. 4b). In the upper section above the upper main fault and the altitude of 1000 m, the YFe 3+ increases drastically and is associated to a decrease in Cr2O3 and Al2O3 contents in chromite (Fig. 4a).

DISCUSSION

Inclusions in chromite: trapped melts or trapped minerals?

There is no consensus about the very origin of the silicate phases included in chromite. The fact that disseminated chromites in pure dunites, i.e. devoid of any impregnation, contain inclusions likewise prevent for the hypothesis of a 'secondary' origin of the inclusions by melt infiltration through cracks and in situ crystallization, and show they were entrapped at the time of the chromite growth.

According to some authors, and in the case of the Maqsad DTZ, those present in chromite from chromitites are former trapped melts (i.e. crystallized in situ after their entrapment) (e.g. [START_REF] Schiano | Primitive basaltic melts included in podiform chromites from the Oman Ophiolite[END_REF][START_REF] Rollinson | Polymineralic inclusions in mantle chromitites from the Oman ophiolite indicate a highly magnesian parental melt[END_REF]. Other authors argue that they are microcrystals already formed in the parent melt or magma at the time they were entrapped (e.g. [START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF]Rospabé et al., 2019b). The homogenization experiments reported in [START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF] have shown that the bulk chemical composition of most inclusions was unlikely the one of any documented natural melt (see also [START_REF] Arai | Comments of the paper "Primitive basaltic melts included in podiform chromites from the Oman ophiolite[END_REF].

Chromites disseminated in the pure dunite of the lower part of the DTZ contain mostly amphibole Journal Pre-Proof and clinopyroxene, mica and diopside inclusions being rather restricted to the chromitite and schlierens. Disseminated chromites in the upper impregnated opx/pl/cpx-bearing-and amph-bearing (± opx/pl/cpx) dunites contain a much diversified mineral content in their inclusions, with in addition to orthopyroxene, plagioclase (mainly anorthitic but one albitic), the more exotic garnet and chlorapatite, as well as most of the mica inclusions (Fig. 1b).

The interstitial minerals between dunites' olivine grains are clearly cumulate minerals crystallized from the last melts that percolated through the mantle-crust transition zone in response to the cooling of the system [START_REF] Rabinowicz | Melt segregation and flow in mantle diapirs below spreading centers: evidence from the Oman ophiolite[END_REF][START_REF] Nicolas | Mantle-crust transition zone and origin of wehrlitic magmas: Evidence from the Oman ophiolite[END_REF][START_REF] Boudier | Nature of the moho transition zone in the Oman ophiolite[END_REF][START_REF] Koga | Petrogenesis of the crust-mantle transition zone and the origin of lower crustal wehrlite in the Oman ophiolite[END_REF][START_REF] Abily | The dunitic mantle-crust transition zone in the Oman ophiolite: Residue of melt-rock interaction, cumulates from high-mgo melts, or both?[END_REF][START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF][START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF]. In the present case, the interstitial orthopyroxene and amphibole were considered as the crystallization product of a lithospheric melt formed at the crust-mantle transition, hybrid between the dry (or water-poor) MORB issued from partial melting within the Maqsad diapir [START_REF] Ceuleneer | Thermal structure of a fossil mantle diapir inferred from the distribution of mafic cumulates[END_REF][START_REF] Benoit | Trace element and isotopic characterization of mafic cumulates in a fossil mantle diapir (Oman ophiolite)[END_REF][START_REF] Python | Nature and distribution of dykes and related melt migration structures in the mantle section of the Oman ophiolite[END_REF] and a hydrated component probably deriving from the deep introduction of seawater or hydrothermal fluids [START_REF] Rospabé | Origin of the dunitic mantle-crust transition zone in the Oman ophiolite: The interplay between percolating magmas and hightemperature hydrous fluids[END_REF]. The correlation between the appearance of such mineral impregnations and the change in the nature of the silicate inclusions in chromite in the higher level of the DTZ, especially the common occurrence of orthopyroxene, suggests than both impregnations and silicate inclusions crystallized from melts sharing common characters, possibly from the same Si-, Na-, trace elements-, and volatile-rich hybrid melt.

Another observation is the clear predominance of monomineralic inclusions relative to polymineralic ones along the cross-section (Fig. 1c). It is more likely explained by the entrapment of tiny crystals rather than by a trapped melt as the complete crystallization of a melt into a single phase must be clearly excluded. As stressed by [START_REF] Rollinson | Polymineralic inclusions in mantle chromitites from the Oman ophiolite indicate a highly magnesian parental melt[END_REF], the surface of an inclusion in a thin section is not representative, and the higher frequency of monomineralic inclusions may be overestimated as it can partly result from the two-dimensional nature of the thin sections, limiting the access to other Journal Pre-Proof companion phases in presence in a given 3-D structure. However, would it be the case, it seems unlikely that 90% of the inclusions observed in thin section would appear monomineralic. This statistically robust result favours the fact that a high proportion of inclusions are actually monomineralic. This is better accounted for if microcrystals formed in the hybrid melt, likely in response of the early stage of the out of equilibrium hybridization process, then were eventually entrapped, mostly as single crystals, during the skeletal growth of chromite [START_REF] Prichard | The structure of and origin of nodular chromite from the Troodos ophiolite, Cyprus, revealed using high-resolution X-ray computed tomography and electron backscatter diffraction[END_REF], consistent with the square to rounded shape of the inclusions (Fig. 2a-b).

In this frame, the amphibole and clinopyroxene that are almost the only minerals observed in inclusion in chromites in pure dunites may represent microcrystals formed during the early stage of hybridization between the MORB and the silica-and water-rich component before the system and the hybrid fluid/melt progressed into a more evolved stage (i.e. more advanced hybridized stage), crystallizing together the more diversified inclusions, chromite and other impregnation minerals upsection (e.g. [START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF]2019a). Since similar inclusions were observed in more concentrated chromitite ore bodies from the Maqsad area [START_REF] Lorand | Silicate and base-metal sulfide inclusions in chromites from the Maqsad area (Oman ophiolite, Gulf of Oman): A model for entrapment[END_REF][START_REF] Leblanc | Chromite crystallization in a multicellular magma flow: Evidence from a chromitite dike in the Oman ophiolite[END_REF][START_REF] Borisova | A new view on the petrogenesis of the Oman ophiolite chromitites from microanalyses of chromite-hosted inclusions[END_REF][START_REF] Rollinson | Polymineralic inclusions in mantle chromitites from the Oman ophiolite indicate a highly magnesian parental melt[END_REF][START_REF] Zagrtdenov | Anatomy of a chromitite dyke in the mantle/crust transition zone of the Oman ophiolite[END_REF]Rospabé et al., 2019b), it is likely that the same mantle dunitization process drove the formation of both the dunites and chromitites. The very details of the dunitechromitite association remain unclear at this stage and this assumption should be investigated further by a more statistical study of the silicate inclusions at the scale of the whole DTZ.

Structural control on the nature of the fluids-melts that formed the DTZ dunites

In our previous study we have reported on the remarkable petrological and geochemical variability along the cross-section studied here in details for its inclusions content (cross-section #3 in Rospabé et al., 2019a). Its lithological stratigraphy (Fig. 1) is consistent with what is observed at the scale of Journal Pre-Proof the whole Maqsad DTZ [START_REF] Rospabé | Etude pétrologique, géochimique et structurale de la zone de transition dunitique dans l'ophiolite d'Oman : Identification des processus pétrogénétiques à l'interface manteau/croûte[END_REF]. The vertical distribution of the silicate inclusions, roughly consistent with the whole rocks and matrix minerals major and trace elements trends, was suspected but needed to be confirmed with a higher analysis density, and the bearing of these variations on the DTZ genesis can now be discussed with an improved degree of confidence.

In addition to the statements discussed above about the nature of the inclusions, the main facts highlighted in this study are: i) the abundance of inclusions in a given sample depends on the host lithology, being globally more abundant in pure dunites in the lower part of the cross-section and decreasing upsection within the impregnated interval (Fig. 1c);

ii) the abundance of silicate inclusions in disseminated chromites locally varies in a coherent way relative to the distribution of fault zones (Fig. 1c), reminiscent to the relation observed between the local higher inclusions content and the presence of magmatic breccias along the stratiform chromitite located about 1.5 km to the SE of the present cross-section (Rospabé et al., 2019b).

In terms of chemical compositions, it can then be observed that: i) vertical evolutions are not random, with successive and alternating trends in most species (Fig. 4), reminiscent to what was already described in the composition of both the dunitic matrix and impregnating minerals of the Maqsad DTZ [START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF]; ii) the vertical chemical evolutions are conditioned by the proximity to the fault zones similarly to the composition of the host dunites (Rospabé et al., 2019a), including for elements considered as immobile during alteration such as the TiO2 content;

iii) approaching fault zones, amphibole in inclusion (the most common phase with clinopyroxene, Fig. 1), is progressively enriched in K (lower Na#), P2O5, F and Cl (Fig. 4b).

The faults and fractures we mapped within the DTZ look to post-date the magmatic activity if only Journal Pre-Proof referring to the alteration mineralogy within fracture planes affecting the host dunite (dominated by serpentine and, in a lesser extent, carbonates). Similarly, faults within the lower crustal section of the Oman ophiolite were described for their role in the important hydrothermal alteration of the layered gabbros [START_REF] Zihlmann | Hydrothermal fault zones in the lower oceanic crust: An example from Wadi Gideah, Samail ophiolite, Oman[END_REF]. On the other hand, strong arguments were provided for the development of higher temperature, syn-magmatic fault zones in the vicinity of the Maqsad area. A N130°E-oriented fault system, parallel to the spreading axis that developed above the Maqsad diapir [START_REF] Pallister | Structure of the sheeted dike complex of the Samail Ophiolite near Ibra, Oman[END_REF][START_REF] Ceuleneer | Evidence for a Paleo-Spreading Center in the Oman Ophiolite: Mantle Structures in the Maqsad Area[END_REF][START_REF] Macleod | Ridge axial segmentation in the Oman ophiolite: Evidence from along-strike variations in the sheeted dyke complex[END_REF], was active at high temperature, early enough during the accretion of the oceanic lithosphere to have disturbed the crystallization of gabbros at the base of the Oman (i.e. buffering of the An content in plagioclase to unusual high values, occurrence of Ti-rich pegmatitic gabbros; [START_REF] Abily | Synmagmatic normal faulting in the lower oceanic crust: Evidence from the Oman ophiolite[END_REF].

Similarly, the fact that i) the vertical chemical evolutions of the inclusions are progressive and continuous along the DTZ, over intervals having a characteristic thickness of about 50 m (Fig. 4) (i.e. not abruptly shifted by the faults), and that ii) it concerns elements for which variations in concentration call for igneous processes such as Ti (in whole rocks, mineral impregnations, chromite, and silicate inclusions), confirms that syn-magmatic faults or fractures developed early within the dunitic mantle-crust transition zone, strongly influenced the nature of the different melt and/or fluid fractions percolating through the DTZ and played an important role on the mobilization of Cr at the mantle-crust transition (Rospabé et al., 2019a).

The higher concentration in phosphorous and halogens, especially Cl, in amphibole inclusions approaching fault zones supports the seawater-derived origin of the fluid fraction involved in the genesis of hybrid melts within the DTZ. This confirms the potential role of fault zones both for extraction of melts and introduction of high temperature hydrothermal fluids (i.e. they make possible the hybridization process, Rospabé et al., 2019a;2019b). Then, at the time of the hybridization the hydrated component/MORB ratio was higher within fault zones than within the massive dunitic Journal Pre-Proof levels, as also attested by the lower TiO2 toward the faults (Fig. 4).

Otherwise, the circulation of very saline fluids (brine-type) and subsequent hydrothermal fluid-rock reaction was documented by the discovery of Cl-rich amphiboles (0.1-5 wt%) in gabbros from the deep lower oceanic crust of the Wadi Tayin massif, Oman (Currin et al., 2018a;2018b). In this way, the Cl-apatite we observed along the cross-section may potentially result from a magmatic to metamorphic process involving such high temperature hydrothermal fluids following the cessation of the magmatic activity in a cooling system.

Another observation is that the abundance of inclusions spatially varies, partly in relation with the presence of faults or fractures (Fig. 1c): higher inclusion contents are roughly observed within the un-faulted dunitic levels, and especially within the chromite-rich interval ~ 750-775 m in altitude, rather than close to the faults (excepting in the lower impregnated level), as opposed to the inclusion distribution relative to magmatic breccias (i.e. frozen stages of fault development) along the neighbouring stratiform chromitite (Rospabé et al., 2019b). This distribution can reflect i) better favoured conditions for the nucleation of the microcrystals within un-faulted levels where the contribution of the MORB is higher, or ii) a faster/earlier formation of the chromite grains relative to the microcrystals close to the faults where the contribution of the hydrated component is higher, or both, preventing significant entrapment of microcrystals. At the scale of the DTZ as a whole, the higher abundance of inclusions in dispersed chromites in pure dunites than in impregnated ones (Fig. 1c) may result from the local thermal state as pure dunite are supposed to have been compacted at higher temperature (i.e. above the crystallization temperature of the interstitial impregnation; [START_REF] Rospabé | Extreme geochemical variability through the dunitic transition zone of the Oman ophiolite: Implications for melt/fluidrock reactions at Moho level beneath oceanic spreading centers[END_REF], the inclusions being the solely crystallization products with chromite in the deeper levels of the DTZ while fractionation from the percolating fluids/melts is also expressed in the impregnations in the shallower levels. This highlights that within the dunitic transition zone, in spite some features are shared between the formation of both dunite and chromitite, local differences Journal Pre-Proof may affect the formation and entrapment conditions of the silicate inclusions in chromite grains. It may reflect the competition between the crystallization of those exotic phases and the massive precipitation of chromite.

CONCLUSIONS

In the DTZ, the inclusion content in the dispersed chromites remarkably correlates with mineral impregnations in the host dunites, both in term of nature and chemical composition. This result first demonstrates that the inclusions in chromite, at least in the case we studied, are not the preserved relics of previous igneous events that happened in a totally different setting (e.g. high pressure processes in the deep mantle), then highlights the need of considering the census and analysis of inclusions in chromite in the systematic survey of the DTZ, in addition to the matrix minerals, to progress in the understanding of the fluid-melt-rock reactions occurring at Moho level below oceanic spreading centres. The development of synmagmatic faults at the mantle-crust transition is proposed i) to occur as early as the dunitization stage implying the re-mobilization of Cr and other elements, ii) to condition the fluid-melt hybridization and fluid-melt-rock reactions in the shallow mantle, and potentially iii) to strongly impact our views concerning the budget of many elements in oceanic spreading settings and the calculation of the global transfers between the mantle, crust and hydrosphere according to deep fluid circulation and hybrid melts genesis. Considering the specific case of halogens, our study suggests that they could be mobilized at very high temperature and not only during low temperature alteration processes (e.g. [START_REF] Kendrick | Halogens and noble gases in Mathematician Ridge meta-gabbros, NE Pacific: implications for oceanic hydrothermal root zones and global volatile cycles[END_REF][START_REF] Chavrit | The contribution of hydrothermally altered ocean crust to the mantle halogen and noble gas Journal Pre-Proof cycles[END_REF][START_REF] Kendrick | Seawater cycled throughout Earth's mantle in partially serpentinized lithosphere[END_REF][START_REF] Kendrick | Halogens in altered ocean crust from the East Pacific Rise (ODP/IODP Hole 1256D)[END_REF].

FIGURE CAPTIONS

Figure 1 :

 1 Figure 1: Vertical distribution of the silicate inclusions in chromite along the studied cross-section.

Figure 2 :

 2 Figure 2: Photomicrographs (reflected light) of a few examples of inclusions in disseminated

Figure 3 :

 3 Figure 3: Major and minor element compositions of the main chromite-hosted silicate inclusions

Figure 4 :

 4 Figure 4: Vertical evolution of the major and minor element compositions of disseminated chromite

Fig. 1

 1 Fig. 1

Fig. 2

 2 Fig. 2

Fig. 3

 3 Fig. 3

Fig. 4

 4 Fig. 4

ACKNOWLEDGMENTS

We gratefully thank F. de Parseval and J.-F. Ména for thin sections realization and J. Langlade for assistance with the EPMA analyses. We are grateful to M. Al Araimi, M. Al Batashi, A. Al-Rajhi, S.

Almusharrafi and other colleagues from the Ministry of Commerce and Industry, Sultanate of Oman, Journal Pre-Proof as well as H. Al Azri, for their constant hospitality. We are grateful to S. Arai and Q. Xiong for their suggestions and very constructive comments as well as to C.-Z. Liu for editorial handling. This work has been financially supported by the Centre National de la Recherche Scientifique-Institut National des Sciences de l'Univers (CNRS-INSU).