Davide Ciucci
email: ciucci@disco.unimib.it

Didier Dubois
email: dubois@irit.fr

Henri Prade
email: prade@irit.fr

Structures of opposition induced by relations The Boolean and the gradual cases

Keywords: Square of opposition, Composition of relations, Fuzzy relation, Multiple-valued logic Mathematics Subject Classfication (2010) 03B52, 68T30

The paper shows that a cube of opposition, a structure that generalizes the square of opposition invented in ancient logic, can be generated from the composition of a binary relation with a subset, by the effect of set complementation on the subset, on the relation, or on the result of the composition. Since the composition of relations is encountered in many areas, the structure of opposition exhibited by the cube of opposition has a universal flavor. In particular, it applies to information processing-oriented settings such as rough set theory, possibility theory, or formal concept analysis. We then discuss how this structure extends to a fuzzy relation and a fuzzy subset, and the graded cube of opposition thus obtained provides an organized view of the different existing compositions of fuzzy relations. The paper concludes by pointing out areas of research where the cube of opposition, or its graded version are of interest.

Introduction

The square of opposition is a structure that exhibits various forms of opposition between four logical statements, and which was originally introduced by Ancient Greek logicians, in relation to the study of syllogisms. This structure is induced by the interplay of an inside and an outside involutive negation operating on the statements. This structure was revisited by medieval logicians before being finally somewhat forgotten with the advent of modern logic more concerned with the formalization of deduction than with issues in knowledge representation related to the use of negation. The square of opposition was tentatively revived in the middle of the XX th century when it was discovered that it can be completed into a hexagon including three squares of opposition, and that the resulting hexagonal structure plays an important role in the organization of concepts [START_REF] Blanché | Structures Intellectuelles. Essai sur l'Organisation Systématique des Concepts[END_REF]. More recently, under the impulse of Jean-Yves Béziau [START_REF] Béziau | New light on the square of oppositions and its nameless corner[END_REF][START_REF] Béziau | The power of the hexagon[END_REF], there has been an increasing surge of interest for the square of opposition and its extensions for applications in philosophical and mathematical logic, linguistics, psychology, semiotics, and artificial intelligence [START_REF]Handbook of Abstracts of the 2nd World Congress on the Square of Opposition[END_REF][START_REF]Handbook of Abstracts of the 3rd World Congress on the Square of Opposition[END_REF][START_REF]Handbook of Abstracts of the 4th World Congress on the Square of Opposition[END_REF].

A cube of opposition can be obtained by duplicating the square of opposition by means of a third negation. This cube is not mentioned very often, and it was rediscovered several times under different guises. It apparently appears for the first time under the form of a cube described by Reichenbach in his modern study of syllogisms [START_REF] Reichenbach | The syllogism revised[END_REF], where he points out that Aristotle himself mentioned the negative oppositions on the back facet of the cube, but did not incorporate them in his theory. The use of the cube for syllogistic reasoning has been recently revisited in [START_REF] Dekker | Not only barbara[END_REF]. The statements associated to the back facet produced by the third negation already appear in [START_REF] Franklin | Some proposed reforms in common logic[END_REF], and in [START_REF] De Morgan | On the structure of the syllogism[END_REF] (where lower case counterparts of the traditional names of the vertices of the square of opposition are introduced for the vertices of the back facet). The same oppositions received a different geometrical representation with an octagon in [START_REF] Keynes | Studies and exercises in formal logic[END_REF] and more recently in [START_REF] Hacker | The octagon of opposition[END_REF]. A similar cube also arises in the study of dual pairs (such as necessity / possibility, universal / existential quantifiers, conjunction / disjunction) in [START_REF] Demey | Algebraic aspects of duality diagrams[END_REF].

The cube of opposition was found again in [START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], where it was noticed that Piaget's group [START_REF] Piaget | Traité de Logique : Essai de Logistique Opératoire[END_REF] of transformations of a compound statement is at work inside the cube. The presence of the cube of opposition structure was then noticed in different formal settings of interest dealing with the processing of information, such as formal concept analysis, possibility theory, abstract argumentation, or rough set theory [START_REF] Amgoud | A formal concept view of abstract argumentation[END_REF][START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF][START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF].

Also recently, Murinová and Novák [START_REF] Murinová | The analysis of the generalized square of opposition[END_REF][START_REF] Murinová | Analysis of generalized square of opposition with intermediate quantifiers[END_REF] have proposed a graded extension of the square of opposition based on Łukasiewicz triangular norm and co-norm, for quantified fuzzy statements. Moreover, they have introduced intermediary layers in the original square of opposition between the top edge linking universally quantified statements, and the bottom edge linking existentially quantified statements, for accommodating soft quantifiers such as "most", or "a few", which then correspond to the vertices of these intermediary layers. Independently of a particular knowledge representation need, suggestions for systematically extending the logical links that should hold between the vertices of a square, of a hexagon, or of a cube of opposition have been provided in [START_REF] Dubois | Gradual structures of oppositions[END_REF]. This is of interest when the vertices may be graded, as for instance in possibility theory. This raises the question of extending the generation process of the cube in the fuzzy (graded) case [START_REF] Zadeh | Fuzzy sets[END_REF].

The paper is organized in the following way. The next section, Section 2, starts with a refresher on the square of opposition insisting on the logical links between the vertices and the dependencies between these links. Section 3 shows how the square of opposition, and then a cube and hexagons, are naturally induced from a relation. Section 4 discusses the graded extensions of these structures on the basis of the links that should exist between the vertices. Section 5 studies the graded cube of opposition induced by a fuzzy relation, which offers a structured organization of the existing fuzzy relational compositions [START_REF] Bandler | Fuzzy power sets and fuzzy implication operators[END_REF][START_REF] Stepnicka | Fuzzy relational compositions based on generalized quantifiers[END_REF]. Sections 2 and 3 are a revised and extended version of the first half of [START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF], while Section 4 elaborates from [START_REF] Dubois | Gradual structures of oppositions[END_REF], and Section 5 is fully new.

The square of opposition: a refresher

Let us start with a refresher on the Aristotelian square of opposition [START_REF] Parsons | The traditional square of opposition[END_REF]. The traditional square involves four logically related statements exhibiting universal or existential quantifications: a statement A of the form "every x is p" is negated by the statement O "some x is not p", while a statement like E "no x is p" is clearly in more general opposition to the first statement (A). These three statements, together with the negation of the last one, namely I "some x is p", give birth to the Aristotelian square of opposition in terms of quantifiers A : ∀x p(x), E : ∀x ¬p(x), I : ∃x p(x), O : ∃x ¬p(x). This square, pictured in Fig. 1, is usually denoted by the letters A, I (affirmative half) and E, O (negative half). The names of the vertices come from a traditional Latin reading: AffIrmo, nEgO).

As can be seen, different relations hold between the vertices. We take them as a basis for a formal definition of the square of opposition. Namely, Definition 1 In a classical square of opposition AIEO, the following relations are supposed to hold: Note that we assume that there are some x for avoiding existential import problems in Fig. 1, and having the condition (b) satisfied. Another well-known instance of this square is in terms of the necessary ( ) and possible (♦) modalities, with the following reading A : p, E : ¬p, I : ♦p, O : ♦¬p, where ♦p = def ¬ ¬p (with p = ⊥, ⊤). Then the entailment from A to I is nothing but the axiom (D) in modal logic, namely p → ♦p.

Fig. 1 Square of opposition

We can describe the links that hold between the vertices of the square using propositional logic, as in, e.g. [START_REF] Miclet | Analogical proportions and square of oppositions[END_REF][START_REF] Smessaert | Logical geometries and information in the square of oppositions[END_REF], where A, I, E, and O are now associated with Boolean variables, i.e., A, I, E, and O are the truth values of statements. Then, the following counterparts of the four above conditions can be easily checked. Let us notice that these conditions are not independent. Indeed, at least the following dependency links hold in any square of opposition.

Structure of opposition induced from a relation and a subset

In this section we describe a very simple process that generates the basic structures of opposition: square, cube, and hexagons. It involves a binary relation and a subset that are composed together.

A relation-based square of opposition

Let us now consider a binary relation R on a Cartesian product X × Y (one may have Y = X). We assume R = ∅. Let xR denote the set {y ∈ Y | (x, y) ∈ R}. We write xRy when (x, y) ∈ R holds, and ¬(xRy) when (x, y) ∈ R. Let R t denote the transposed relation, defined by xR t y if and only if yRx, and yR t will be equivalently denoted as Ry = {x ∈ X | (x, y) ∈ R}.

Moreover, we assume that ∀x, xR = ∅, which means that the relation R is serial, namely ∀x, ∃y such that xRy; this is also referred to in the following as the X-normalization condition. In the same way R t is also supposed to be serial, i.e., ∀y, Ry = ∅ (Y -normalization). We further assume that the complementary relation R (xRy iff ¬(xRy)), and its transpose are also serial, i.e., ∀x, xR = Y and ∀y, Ry = X. These conditions enforce a non trivial relation between X and Y . In the following, set complementations will be denoted by means of overbars.

Let S be a subset of Y . We assume S = ∅ and S = Y . The relation R and the subset S, also considering its complement S, give birth to the two following subsets of X, namely the (left) images of S and S by R

R(S) = {x ∈ X | ∃s ∈ S, xRs} = {x ∈ X | S ∩ xR = ∅} = s∈S Rs (1)
R(S) = {x ∈ X | ∃s ∈ S,
The four subsets thus defined can be nicely organized into a square of opposition, see Fig. 2. Indeed, it can be checked that the set counterparts of the relations (a)-(d) of Proposition 1 thanks to the X-normalization condition ∀x, xR = ∅. These inclusions are represented by vertical arrows in Fig. 2; (c) R(S) ∩ R(S) = ∅ ; this empty intersection corresponds to the thick line in Fig. 2, and one may have

R(S) ∪ R(S) = Y ; (d) R(S) ∪ R(S) = X;
this full union corresponds to the double thin line in Fig. 2, and one may have R(S) ∩ R(S) = ∅.

Conditions (c)-(d) hold also thanks to the X-normalization of R.

Note that this fits with a regular modal logic reading of this square where R is viewed as an accessibility relation defined on X × X, and S as the set of models of a proposition p. Indeed, p (resp. ♦p) is true in world x means that p is true at every (resp. at some) possible world accessible from x; this corresponds to R(S) (resp. R(S)) which is the set of worlds where p (resp. ♦p) is true. It is easy to check axiom K is valid.

Proposition 3

The relational square satisfies axiom K: ( (p → q) → ( p → q)).

Proof

(p → q) → ( p → q) writes in set theoretic terms

R(S → T ) → (R(S) → R(T )) = Y
where S → T stands for S ∪ T . Developing, it comes

R(S ∩ T ) ∪ R(S) ∪ R(T ) = Y It means R(T ) ⊆ R(S ∩ T ) ∪ R(S), but note that R(S ∩ T ) ∪ R(S) = R((S ∩ T ) ∪ S) = R(T ∪ S)
, so that it holds indeed.

Moreover, the entailment from A to I is the axiom (D) of modal logic which is known to require serial accessibility relations [START_REF] Chellas | Modal Logic: An Introduction[END_REF]. So the logical framework of the relational square of opposition seems to be that of the logic KD.

The relational cube

Let us also consider the complementary relation R, namely xRy if and only if ¬(xRy). We further assume that R = ∅ (i.e., R = X × Y ). Moreover we have also assumed the Xnormalization of R, i.e., ∀x, ∃y ¬(xRy). In the same way as previously, we get four other subsets of X from R. Namely,

R(S) = {x ∈ X | ∃s ∈ S, ¬(xRs)} = {x ∈ X | S ∪ xR = X} = s∈S Rs (3)
The eight subsets involving R and its complement can be organized into a cube of opposition [START_REF] Ciucci | Rough sets, formal concept analysis, and their structures of opposition extended abstract[END_REF][START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF] (see Fig. 3). Similar cubes have been recently shown to extend the traditional square of opposition in terms of quantifiers [START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], or in the particular setting of abstract argumentation (for the complement of the attack relation) [START_REF] Amgoud | A formal concept view of abstract argumentation[END_REF]. As can be seen, the front facet of the cube in Fig. 3 is nothing but the square in Fig. 2, and the back facet is a similar square induced by R. Its vertices are denoted by lower case letters a, i, e, and o [START_REF] De Morgan | On the structure of the syllogism[END_REF][START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF]. Neither the top and bottom facets, nor the side facets are squares of opposition in the above sense since condition (a) is violated: Diagonals do not link complements in these squares. More precisely, in top and bottom squares, diagonals change R into R and vice versa; there is no counterpart of condition (b), in the top square condition (c) holds for the pairs of subsets associated with vertices A-E and with vertices a-e, while condition (d) fails (top square), and conversely in the bottom square, condition (d) holds for the pairs of subsets associated with vertices I-O and with vertices i-o while condition (c) fails. Moreover, the top and bottom facets exhibit other empty intersection relationships and full union relationships respectively. Indeed in the top facet, e.g.

R(S)∩R(S) = ∅, since R(S) = {x ∈ X | S ⊆ xR} and R(S) = {x ∈ X | S ∩ xR = ∅}. Similarly in the bottom facet, e.g. R(S) ∪ R(S) = X, since R(S) = {x ∈ X|∃s ∈ S, ¬(xRs)} and R(S) = {x ∈ X|∃s ∈ S,
xRs}. This is pictured in Fig. 4 (in order not to overload Fig. 3).

For side facets, condition (b) clearly holds, while both conditions (c)-(d) fail. In side facets, vertices linked by diagonals are exchanged by changing R into R (and vice versa) and by applying the overall complementation. These diagonals express set inclusions: R(S) = {x ∈ X|∀s ∈ S, ¬(xRs)} ⊆ {x ∈ X|∃s ∈ S, ¬(xRs)} = R(S). In the same way, we have R(S) ⊆ R(S), R(S) ⊆ R(S), and R(S) ⊆ R(S), as pictured in Fig. 5 for the left side. Mind that while diagonals in front and back facets express complementations, they express inclusions in side facets, empty intersections in top facet, and full union in bottom facets. As can be seen, the conditions associated with A, a, I, and i can be written respectively while O,o,E, and e are respectively associated with their negation. Note that although the vertices of these side squares present some forms of structural opposition, e.g. A and a, these squares strongly differ from the original square of opposition. Moreover, the conditions associated to the vertices involve all possible ways of comparing the subsets xR and S possibly using their complements. Indeed the conditions xR ⊆ S, S ∩ xR = ∅, S ⊆ xR, and S ∪ xR = X express the four possible inclusion relations of xR wrt S or S and define distinct subsets of X. Thus, the 4 subsets R(S), R(S), R(S), and R(S) (or their complements R(S), R(S), R(S), and R(S)) constitute distinct pieces of information in the sense that one cannot be deduced from the others [START_REF] Dubois | A possibility-theoretic view of formal concept analysis[END_REF].

xR ∩ S = ∅, xR ∩ S = ∅, xR ∩ S = ∅, xR ∩ S = ∅,
In the cube of opposition, three negations are at work, the usual outside one, and two inside ones respectively applying to the relation and to the subset -this gives birth to the eight vertices of the cube -while in the front and back squares (but also in the top and bottom squares) only two negations are at work. Besides, it is obvious that a similar cube can be built for the transpose relation R t and a subset T ⊆ X, then inducing eight other remarkable subsets, now, in Y . This leads us to assume the X-normalizations of R t and R t (= R t ), which is nothing but the Y-normalizations of R and R (∀y, ∃t, tRy, and ∀y, ∃t ′ , ¬(t ′ Ry)), as already announced. In other words, we have to assume the seriality of R, R t , R, and R t . Top and bottom facets :

(i) a and E cannot be true together, but may be false together;

(j) the same for A and e;

(k) i and O cannot be false together, but may be true together;

(l) the same for I and o.

The 8 subsets corresponding to the vertices of the cube of opposition can receive remarkable interpretations in the settings of possibility theory (assuming no intermediary degree between fully possible and impossible) [START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], formal concept analysis [START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF][START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], and rough set theory [START_REF] Ciucci | Oppositions in rough set theory[END_REF][START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF]. The front facet of the cube corresponds to the core of rough set theory, while formal concept analysis operators are on the left-hand side face, and the set functions of possibility theory are associated with the eight vertices of the cube.

It is also interesting to complete the different squares corresponding to the facets of the cube into hexagons, as we are going to see.

From squares to hexagons

As proposed and advocated by Blanché [START_REF] Blanché | Sur l'opposition des concepts[END_REF][START_REF] Blanché | Structures Intellectuelles. Essai sur l'Organisation Systématique des Concepts[END_REF], it is always possible to complete a classical square of opposition into a hexagon: A,E, and Y always give birth to a hexagon [START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], which is made of three squares of opposition: AEOI, AYOU, and EYIU, as in Fig. 6. The interest of this hexagonal construct has been Fig. [START_REF]Handbook of Abstracts of the 3rd World Congress on the Square of Opposition[END_REF] Hexagon associated with the front facet of the cube rediscovered and advocated again by Béziau [START_REF] Béziau | New light on the square of oppositions and its nameless corner[END_REF] in recent years in particular for solving delicate questions in paraconsistent logic modeling.

Applying this idea to the front facet of the cube of opposition induced by a relation and a subset, we obtain the hexagon of Fig. 6, associated with the tri-partition

{R(S), R(S), R(S) ∩ R(S)}. Note that indeed R(S) = R(S) ∪ (R(S) ∩ R(S)) (since R(S) ⊇ R(S)). Similarly, R(S) = R(S) ∪ (R(S) ∩ R(S)
). In Fig. 6, arrows (→) indicate set inclusions (⊆). A similar hexagon is associated with the back facet, changing R into R.

Another type of hexagon can be associated with side facets. The one corresponding to the left-hand side facet is pictured in Fig. 7. Now, not only the arrows of the sides of the hexagon correspond to set inclusions, but also the diagonals (oriented downwards). Indeed R(S) ⊆ R(S) and R(S) ⊆ R(S). Moreover, since R(S) ⊆ R(S) and using the inclusions corresponding to the vertical edges of the cube, we get One may wonder if one can build useful hexagons from the bottom and top squares of the cube. It is less clear, as shown by the following discussion. Indeed, if we consider the four subsets involved in the bottom square, namely R(S), R(S), R(S) and R(S) (the top square has their complements as vertices), they are weakly related through R(S) ∪ R(S) = X, R(S)∪R(S) = X, R(S)∪R(S) = X and R(S)∪R(S) = X. Still R(S)∪R(S) or R(S)∪R(S) (or their complements in the top square, R(S)∩R(S) or R(S)∩R(S)) are compound subsets that may be of interest for some particular understanding of relation R. Note that in general, R(S) ∪ R(S) = X and R(S) ∪ R(S) = X. Note also that similar combinations changing ∩ into ∪ and vice versa, namely R(S) ∪ R(S) and R(S) ∩ R(S) already in the hexagons associated with the side facets of the cube, while R(S) ∩ R(S) and R(S) ∩ R(S) are the Y-vertices of the hexagons associated with the front and back facets of the cube of opposition.

R(S) ∪ R(S) ⊆ R(S) ∩ R(S),
Besides, it would be also possible to complete the top facet into yet another type of hexagon by taking the complements of R(S) ∪ R(S) or of R(S) ∪ R(S), which clearly have empty intersections with the subsets attached to vertices A and a and to vertices E and e respectively. However, the intersection of the two resulting subsets, namely R(S)∩R(S) and R(S) ∩ R(S) is not necessarily empty. A dual construct could be proposed for the bottom facet.

Boolean closure of the cube

In search of the hexagons arising from the different facets of the cube, several combinations of the vertices have been considered, obtaining new subsets of the universe from R and S. One may push this procedure further, and study the Boolean closure of the cube, that is all the non-trivial (i.e., different from the empty set and the universe) subsets that can be obtained from the starting eight ones using standard set operations. We notice that the hexagon is the Boolean closure of the square.

As we may expect, the Boolean closure of the cube is more complicated to compute than the Boolean closure of the square. At least one may wonder if it contains other cubes of the same kind. Indeed, one may consider the 3 potential cubes made by pairing the 3 squares of the hexagon of the front facet with the 3 corresponding squares of the hexagon of the back facet. However, the 2 additional potential cubes thus obtained do not make cubes of opposition in the above sense, since the expected relations on the side facets do not hold; for instance, A does not entail u, where u = a ∨ e.

Gradual structure of opposition

We have examined the abstract logical structure underlying the classical square of opposition and the derived structures. The statements associated to the vertices were supposed to be true or false, and the membership in the counterpart in terms of relation and subset was all-or-nothing. We now discuss how these structures could be generalized when truth, or membership, become graded. This is of interest since various settings, for which structures of opposition are relevant, are naturally graded, such as possibility theory.

The graded square of opposition

Let us now associate a degree in Then using n(s) = 1s together with * = min, we get a Kleene system leading to a square of opposition. An instantiation of such a graded square of opposition is provided by the main set functions in possibility theory applied to classical events [START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF].

Example 1 Given a (finite) set , and a possibility distribution π : → [0, 1], assumed to be normalized (i.e., ∃ω ∈ , π(ω) = 1), let us denote by (A) = max ω∈A π(ω) the possibility degree of a proposition with set of models A, and by N (A) = 1 -(A) the conjugate necessity degree. We can instantiate the gradual square of opposition by letting α = N (A), ǫ = N (A), ι = (A), o = (A). Then (a) is nothing but the duality between possibility and necessity degrees. Conditions (b), (c) and (d) hold as well, since respectively we have in possibility theory that (A) < 1 entails N (A) = 0 (or equivalently N (A) > 0 entails (A) = 1), min(N(A), N(A)) = 0, and max( (A), (A)) = 1.

Łukasiewicz systems If we take Łukasiewicz triangular norm [START_REF] Klement | Triangular Norms[END_REF][START_REF] Schweizer | Probabilistic Metric Spaces[END_REF], namely, max(0, •+ • -1) for the conjunction * , the conditions write

(a) α = n(o); ǫ = n(ι); (b) n(ι) ≤ 1 -α and n(o) ≤ 1 -ǫ; (c) α + ǫ ≤ 1; (d) n(ι) + n(o) ≤ 1.
Choosing again n(s) = 1s simplifies the above conditions into

(a) α = 1 -o; ǫ = 1 -ι; (b) α ≤ ι and ǫ ≤ o; (c) α + ǫ ≤ 1; (d) ι + o ≥ 1.
Such a graded extension of the square of opposition is also a remarkable instance of another point of view we may think of for extending the square of opposition with grades. Namely, starting only from a symmetrical conjunction * , we may both build the implication → and the negation ν by residuation. Namely, s → t = max{x ∈ [0, 1] | s * x ≤ t}, and ν(s) = s → 0. Taking * = min, it yields Gödel implication s → t = 1 if s ≤ t and s → t = t otherwise. Unfortunately in this case, the negation obtained, namely ν(s) = 1 if s = 0, and ν(s) = 0 if s > 0, is no longer many-valued, nor involutive, and then the symmetries of the square of opposition are completely lost, that is conditions (a)-(d) do not hold. However, starting with * = max(0, • + • -1), we obtain Łukasiewicz implication s → t = min(1, 1s + t), and then ν(s) = 1s as a negation, which is involutive. This again corresponds to the above system of constraints.

Note that the latter system of constraints is weaker than the Kleene system, since we have the following entailments

(b) min(α, 1 -ι) = 0 entails α ≤ ι and min(ǫ, n(o)) = 0 entails ǫ ≤ o; (c) min(α, ǫ) = 0 entails α + ǫ ≤ 1; (d) max(ι, o) = 1 entails ι + o ≥ 1.
An instantiation of the above weaker system is provided by possibility theory where possibility and necessity measures are extended to fuzzy events, as explained below.

Example 2 (Example 1 continued) The possibility of a fuzzy event A [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF], represented by its membership function µ A : → [0, 1], is defined by (A) = max ω∈ min(µ A (ω), π(ω)), and its necessity by N (A) = 1 -(A). Letting α = N (A), ǫ = N (A), ι = (A), o = (A) again, (a) always expresses the duality between possibility and necessity degrees, while condition (b), (c) and (d) still hold in a weaker way with respect to the case of classical events, since we have for fuzzy events N (A) ≤ (A), N (A) + N (A) ≤ 1, and (A) + (A) ≥ 1 respectively.

More generally, we have the following result for nilpotent triangular norms [START_REF] Klement | Triangular Norms[END_REF][START_REF] Schweizer | Probabilistic Metric Spaces[END_REF] (i.e., triangular norms * that are continuous and such that for each x in the open interval (0, 1) there is a natural number n such that x * • • • * x (n times) equals 0). * = max(0, • + • -1) is the prototype of nilpotent triangular norms. Zadeh systems Note that the preservation of the whole structure of dependencies in the square of opposition relies on the use of the strong implication associated with the conjunction, for expressing conditions (b). Instead, we may prefer to choose the implication independently from * , and use for (b) weaker conditions such as the ones obtained with residuated implications, namely α ≤ ι and ǫ ≤ o. These weaker (b)-conditions are of interest for dealing for instance with fuzzy sets whose inclusion is defined pointwisely by such inequalities. As an illustration, take n(s) = 1s, * = min, together with the conditions α ≤ ι and ǫ ≤ o, we obtain the following extension of the square of opposition, which may be of interest, but where the entailment of (c) and (d) from (a) and (b) no longer holds:

(a) α = 1 -o; ǫ = 1 -ι; (b) α ≤ ι and ǫ ≤ o; (c) min(α, ǫ) = 0; (d) max(ι, o) = 1;
Combining squares In all the previous examples, we always considered triangular norms as conjunctions, even if associativity, and increasingness played no role. A situation where these two properties are required is when considering the idea of combining squares of opposition as recently proposed in [START_REF] Pizzi | Contingency Logics and Modal Squares of Opposition[END_REF]. A, I, E, O andA

Proposition 5 Given two squares of opposition

′ , I ′ , E ′ , O ′ , it can be checked that A ∧ A ′ , I ∨ I ′ , E ∧ E ′ , O ∨ O ′
make another square of opposition satisfying the four characteristic conditions. This fact extends to the graded case using the same conjunction * and disjunction ⊕ for the combination of the squares.

Proof With obvious notations, the following can be easily checked:

(a) α = n(o) and α ′ = n(o ′ ) entail α * α ′ = n(o ⊕ o ′ ). Similarly, ǫ = n(ι) and ǫ ′ = n(ι ′ ) entail ǫ * ǫ ′ = n(ι ⊕ ι ′ ). (b) α * n(ι) = 0 and α ′ * n(ι ′ ) = 0 entail α * α ′ * n(ι) * n(ι ′ ) = 0 using commutativity and associativity, i.e., (α * α ′ ) * n(ι ⊕ ι ′ ) = 0. Similarly, ǫ * n(o) = 0 and ǫ ′ * n(o ′ ) = 0 entail (ǫ * ǫ ′ ) * n(o ⊕ o) = 0. (c) α * ǫ = 0 and α ′ * ǫ ′ = 0 entail (α * α ′ ) * (ǫ * ǫ ′ ) = 0 (using associativity). (d) n(ι) * n(o) = 0 and n(ι ′ ) * n(o ′ ) = 0 entail n(ι ⊕ ι ′ ) * n(o ⊕ o ′ ) = 0.
The combination of the squares then remains associative thanks to the associativity of triangular norms. Besides, note that in case we use the weak form of (b), namely α ≤ ι and ǫ ≤ o, then α ≤ ι and α ′ ≤ ι ′ entail, from the increasingness of * , that α * α ′ ≤ ι * ι ′ (and we have a similar entailment for the second condition). An example where this combination of squares make sense is again exemplified using possibility theory.

Example 3 (Examples 1 and 2 continued) If two squares of opposition correspond respectively to possibility and necessity functions i , N i , i = 1, 2 based on two possibility distributions π 1 and π 2 , then the two corresponding squares of opposition can be combined into one square by setting α = min(N

1 (A), N 2 (A)), ǫ = min(N 1 (A), N 2 (A)), ι = max( 1 (A), 2 (A)), o = max( 1 (A), 2 (A))
, which yields a Kleene square of opposition for non fuzzy events. The reason is that max( 1 (A), 2 (A)) = max ω∈A max(π 1 (ω), π 2 (ω)) is the possibility function induced by the possibility distribution max(π 1 , π 2 ) [START_REF] Dubois | Aggregation of possibility measures[END_REF]. For fuzzy events, only the weak form of (b) will be preserved.

Graded cube of opposition

As for the square of opposition, graded extensions can be proposed for the cube. Let us suppose that the front and back of the cube are square of opposition and α, ι, ǫ, o, and α ′ , ι ′ , ǫ ′ , o ′ be the grades associated to vertices A, I, E, O anda, i, e, o. Then, we can define a graded cube of opposition in the following way. Then, the constraints associated with the side facets (e)-(h) and with the top and bottom facets (i)-(l) read as:

(e) α * n(ι ′ ) = 0 (f) α ′ * n(ι) = 0 (g) ǫ ′ * n(o) = 0 (h) ǫ * n(o ′ ) = 0. (i) α ′ * ǫ = 0 (j) α * ǫ ′ = 0 (k) n(ι ′ ) * n(o) = 0 (l) n(ι) * n(o ′ ) = 0.
Then, it can be straightforwardly checked that given conditions (a)- (b)-(c)-(d), conditions (e)-(f)-(g)-(h) are equivalent to conditions (i)-(j)-(k)-(l). Moreover, one may think of using the weak view of implication, which amounts to writing for the front and the back facets (b') in place of (b):

(b') α ≤ ι, ǫ ≤ o and α ′ ≤ ι ′ , ǫ ′ ≤ o ′ ;
and for the side facets:

(e') α ≤ ι ′ ; (f') α ′ ≤ ι; (g') ǫ ′ ≤ o; (h') ǫ ≤ o ′ .
In the general case, the top (equiv., bottom) and side constraints cannot be derived from the conditions on the square neither one from the other. For instance, if we consider the drastic triangular norm as the conjunction * , namely s * t = s if t = 1, s * t = t if s = 1, and s * t = 0 otherwise, it is easy to find values that satisfy the top / bottom constraints and not the side ones. Conversely, using * = min, we easily get that side facets can be satisfied without satisfying the top / bottom ones.

If we put more constraints on the conjunction, we have the following result.

Proposition 6 If * is a left-continuous triangular norm and its residuated implication → * has contrapositive symmetry (that is, x → * y = n(y) → * n(x)), then the top and bottom conditions are equivalent to the side ones.

Proof From α ′ * ǫ = 0, we get α ′ * n(ι) = 0 which, given the hypothesis, holds iff α ′ ≤ ι (see [START_REF] Fodor | Contrapositive symmetry of fuzzy implications[END_REF]). The other conditions are obtained in a similar way.

This result should not come as a surprise since the prototypical case of residuated implication having contrapositive symmetry is Łukasiewicz implication, which is also a strong implication with respect to * = max(0, • + • -1).

Weaker results can be proved for other triangular norms. For instance,

Proposition 7 In case of * = min we have that top / bottom conditions imply the (e')-(f')-(g')-(h') version of side ones.

Proof Condition min(α ′ , ǫ) = 0 implies α ′ = 0 or ǫ = 0 and consequently α ′ ≤ ι = n(ǫ).

On the other hand, α ′ can be less or equal to ι with α = 0 and ι = 1.

The converse of the above proposition does not generally hold. Such a graded cube can receive different instantiations. One would be in terms of fuzzy relations as discussed in the last section of this paper. Another is in terms of (graded) possibility theory [START_REF] Dubois | Possibility theory: Qualitative and quantitative aspects[END_REF], as briefly indicated below.

Example 4 (Examples 1, 2 and 3 continued) Assuming that the normalized possibility distribution π : → [0, 1], is also such that 1π is normalized (i.e., ∃ω ∈ , π(ω) = 0), let us denote by (A) = min ω∈A π(ω) the strong / guaranteed possibility degree of a proposition with set of models A, and by ∇(A) = 1 -(A) the conjugate degree. We can instantiate the gradual square of opposition by letting α ′ = (A), ǫ ′ = (A), ι ′ = ∇(A), o ′ = ∇(A). Thanks to the duality between and ∇, and normalization of 1π , it can be checked that α ′ , ǫ ′ , ι ′ , o ′ makes a square of opposition for * = min, and n(s) = 1s. Since (A) ≤ (A) and ∇(A) ≤ N (A), the constraints of the side facets hold under the form (e')-(f')-(g')-(h').

The gradual hexagon

Now, let us investigate the graded counterpart of the hexagon. We will denote as ν and γ the degrees of truth associated to U and Y (since they both collapse into the letter upsilon in Greek). Then, we should have

ν = α ⊕ ǫ and γ = ι * o (5)
If we want to preserve AYOU as a square of opposition, we should have that (a) ν = n(ǫ). This is true once condition ( 5) is satisfied. Indeed,

ν = α ⊕ ǫ = n(ι) ⊕ n(o) = n(ι * o) = n(γ ). (b) A entails U and Y entails O, that is α ≤ ν and γ ≤ o 1 .
Using again condition [START_REF]Handbook of Abstracts of the 2nd World Congress on the Square of Opposition[END_REF], this means α ≤ α ⊕ ǫ and ι * a ≤ o, which is true for any choice of monotonic conjunction * and disjunction ⊕, and in particular for all triangular norms and triangular conorms. (c) α * γ = 0. This condition, generally, does not follow from the previous ones, so that we should impose it. (d) n(o) * n(ν) = 0. This condition is equivalent to the previous one.

Similarly, in the case of the square EYIU, we should impose that Y entails I , E entails U and ǫ * γ = 0 (or equivalently, n(i) * n(ν) = 0).

Proposition 8

In the case of nilpotent triangular norms such as * = max(0, • + • -1), condition [START_REF]Handbook of Abstracts of the 2nd World Congress on the Square of Opposition[END_REF] alone ensures that all the three squares AEOI, AYOU, and EYIU satisfy conditions (a)-(d) as well as the set of dependencies among them.

Proof α * γ = 0 leads to the constraint n(α * n(ν)) = 1, which is equivalent to n(α * n(α ⊕ ǫ)) = 1, i.e., we have, α * (n(α) * n(ǫ)) = 0. Assuming that * is a triangular norm, and applying associativity we get (α * n(α)) * n(ǫ) = 0. Since it should hold for any ǫ, it leads to require that α * n(α) = 0 for any α, which amounts to saying that * is a nilpotent triangular norm, typically * is the Łukasiewicz triangular norm s * t = max(0, s + t -1). Then, we already know, by Proposition 4 that with a nilpotent triangular norm and the four constraints (a)-(b)-(c)-(d), all the dependencies are satisfied for the three squares AEOI, AYOU, and EYIU. We thus get a perfect graded extension of the hexagon. Summarizing, -in order to have a hexagonal structure with three squares of opposition, it is sufficient to satisfy condition (5), condition (c) (for the three squares) and the fact that * and ⊕ are dual triangular norms; -condition (5) and a nilpotent triangular norm (with its dual co-norm) are sufficient to satisfy condition (a)-(d) for all the three squares as well as their dependencies.

Conversely, given a fuzzy partition in the sense of Ruspini [START_REF] Ruspini | A new approach to clustering[END_REF], we can generate a hexagon. This result could be generalized to any nilpotent triangular norm using a partition of the form ϕ(α) + ϕ(ǫ) + ϕ(γ ) = 1 where ϕ is the additive generator of the triangular norm.

Note also that the constraint α + ǫ + γ = 1 can be weakened into α + ǫ + γ ≤ 1 without any harm for the triple square-of-opposition structure. This should not come as a surprise since the mutual exclusiveness of the three situations A, E, Y is the key condition for having the hexagon in the non-graded case, while there is no covering condition. We notice that these two different hexagons are named, respectively, strong and weak in [START_REF] Pellissier | setting" n-opposition[END_REF]. The context there is different, since based on modal logic, but the interpretation is similar. Indeed, a weak hexagon is such that the disjunction of the three vertices is not necessarily true whereas it is true in a strong hexagon and in both cases the conjunction of any two (out of three) vertices is false.

Finally, let us consider the hexagon built on the side of the cube (see Fig. 7). With respect to the constraints holding on the cube, we should add the ones concerning U and Y:

α ≤ γ , α ≤ ν, α ′ ≤ γ , α ′ ≤ ν, γ ≤ ι, γ ≤ ι ′ , ν ≤ γ .
From this list, we notice that it must hold that α ≤ 0.5. This is due to the fact that α ≤ ν and α ≤ n(ν) = γ . The same holds for α ′ . Then, since it should hold that ν ≤ γ = n(ν), we get that ν ≤ 0.5 and γ ≥ 0.5 and consequently ι, ι ′ ≥ 0.5. That is, the upper part of the hexagon is made of all values less or equal to 0.5 and the lower part of all values greater or equal to 0.5.

Structures of opposition and the composition of two relations

In this section, we consider, as basic bricks to build our structures of opposition, two relations and the different possibilities to combine them. This is a well-known topic both in the Boolean and fuzzy settings [START_REF] Bandler | Fuzzy power sets and fuzzy implication operators[END_REF][START_REF] De Baets | Fuzzy relational compositions[END_REF][START_REF] Kohout | Relational-product architectures for information processing[END_REF]]. We will discuss how these approaches fit into our framework.

The Boolean case

Let us suppose that R ⊆ (X × Y ) and S ⊆ (Y × Z), we want to build new relations on X × Z by a composition of R, S. In order to build a square of opposition, and in particular to guarantee that the entailments from top to bottom of the square hold, we have to take care of the normalization conditions. That is, R, S and their transpose should be serial and consequently xR, Ry, yS and Sz cannot be empty. Assuming this condition we have that the four corners of the square read as follows. We notice that (A) is the Bandler and Kohout [START_REF] Kohout | Relational-product architectures for information processing[END_REF] subproduct, (I) is the basic composition of relations, and clearly (O) and (E) are their negations. To be more precise, (A) is the De Baets and Kerre [START_REF] De Baets | Fuzzy relational compositions[END_REF] version of the subproduct due to the presence of the normalization conditions, which are not considered by Bandler and Kohout.

Example 5 Let us consider the example 2.1 "medical diagnosis" taken from [START_REF] De Baets | Fuzzy relational compositions[END_REF]. So, R is a relation between a set of patients X and symptoms Y and S a relation between symptoms and illnesses Z. Then, we have the following interpretation: x(I)z iff the patient x has at least one symptom associated to the illness z; x(A)z iff all the symptoms of x are symptoms of the illness z. Now, the negations of these two relations can be interpreted as: x(E)z iff x has none of the symptoms of z; x(O)z iff x has some symptom not related to the illness z.

Similarly, and with the normalization conditions applied to the negation of R and S, the back square of the cube is made by the four corners In this case, we have that (a) is the De Baets and Kerre reading of Bandler and Kohout superproduct and (o) its negation. On the other hand, (e) and its negation (i) seem to be new ways to compose relations. The corner (e) expresses the idea that x and z are in relation if they are in connection (through R and S) to the rest of the world.

Example 6 (Example 5 continued) Coming back to the example of medical diagnosis, we have the following interpretations: x(a)z iff the patient x has all the symptoms of the illness z (and perhaps some more); x(o)y iff there exists a symptom of the illness z that the patient x does not have; x(e)z iff the symptoms possessed by x plus the symptoms characterizing z cover the whole universe of symptoms; finally x(i)z is the negation of this last situation: there exists a symptom not possessed by x and not characterizing z. Now, let us turn our attention to the hexagons. The one built on the front face adds the two corners to the traditional square:

(U) R • S ∪ R • S = {(x, z)|xR ⊆ Sz or xR ∩ Sz = ∅} (Y) R • S ∩ R • S = {(x, z)|xR ∩ Sz = ∅ and xR ∩ Sz = ∅}
That is, (U) represents the case where x, z are in relation if z is connected to all or none of the elements connected to x. On the other hand, (Y) implies that there exists at least one element related to x and to z and another one related to x and not to z.

Example 7 (Examples 5 and 6 continued) In terms of Example 5, (U) represents the situation where the symptoms of a patient x are all or none of the symptoms of an illness z. The corner (Y) represents the case of a patient with some symptoms of an illness z and some symptoms not from z.

If we consider the hexagons, build on the left side (see Fig. 7), the new corners introduce new operators, which however do not seem to have an interesting interpretation with respect to the composition of relations. They are the union of (A) and (a), i.e., the union of super and sub-product, and the intersection of (I) (the standard composition) and (i). On the other hand, in this case one of the possible hexagons built on the top deserves some interest, since the conjunction of (a) and (A) defines the square-product, the fourth and last operation introduced by Bandler and Kohout. This particular hexagon is completed by the new operator (e) ∪ (E).

The gradual case

By giving a reading of the composition of relations in terms of characteristic functions, De Baets and Kerre [START_REF] De Baets | Fuzzy relational compositions[END_REF] give two different generalizations of Bandler and Kohout operations. We start from them in order to give a generalization of the cube to the gradual case.

Let * be a triangular norm, R and S fuzzy relations, → a fuzzy implication. Then, the standard composition of relations, corresponding to the corner (I) is:

(R • S)(x, z) = sup y∈Y R(x, y) * S(y, z)
By its negation we obtain (E), using n(s) = 1s:

(R • S)(x, z) = 1 -sup y∈Y R(x, y) * S(y, z) = inf y∈Y (1 -R(x, y)) ⊕ (1 -S(y, z))
The corner (A) corresponds to the fuzzified subproduct. The two (different) definitions given by De Baets and Kerre are: Assuming that R and S are serial means that the two last terms in (A ′ ) are equal to 1. Moreover as the sup- * composition preserves seriality, the second term in (A) is also 1. So, for serial fuzzy relations the extra terms in (A) and (A ′ ) can be deleted, (A)=(A ′ ) and reads inf y∈Y R(x, y) → S(y, z).

Under the seriality assumption, (A) does reduce to R • S in the crisp case since then R • S ⊆ R • S. It is also true for (A ′ ) since sup y∈Y (R(x, y) * S(y, z)) ≤ sup y∈Y min(R(x, y), S(y, z)) ≤ min(sup y∈Y R(x, y), sup y∈Y S(y, z)). The negations of the two above expressions correspond to the corner (O ′ ):

(O) max sup y∈Y (1 -(R(x, y) → S(y, z))), inf y∈Y (1 -R(x, y)) ⊕ (1 -S(y, z)) (O ′ ) max sup y∈Y (1 -(R(x, y) → S(y, z))), inf y∈Y 1 -R(x, y), inf y∈Y 1 -S(y, z)
Again, if the fuzzy relations are serial, the second term in (O) and the second and third terms in (O ′ ) can be dropped. Now, let us see if the requirements to get a gradual square of opposition hold. We assume that we use the weak entailment since as already said, it fits with fuzzy set inclusion. By definition we have that (A) and (O), (A ′ ) and (O ′ ), (E) and (I) are complements. Then, it is clear that (A) is less than or equal to (I) and (E) is less than or equal to (O). On the other hand, it may happen that (A ′ ) is greater than (I) and similarly (E) is greater than (O ′ ). For instance, if for all y ∈ Y we have: R(x, y) = 1 3 , S(y, z) = 1 2 and we consider the product triangular norm and its residuum, we get that the degree of (A ′ ) is 1 3 and the degree of (I) is 1 6 . Finally, the constraint α * ǫ = 0 requires that k * (1k) = 0 (where k = sup y∈Y R(x, y) * S(y, z)). Again, this holds if * is a nilpotent triangular norm (see Proposition 4). As a conclusion, (A), (I), (E) and (O) form a square of opposition once * is a nilpotent triangular norm.

Instead of computing (E) and (O) as negations of (I) and (A), we can obtain them directly as a fuzzification of the corresponding Boolean formulae. In the case of (E) we can proceed as for (A) by substituting S(y, z) with its negation 1 -S(y, z). So we get two possible definitions Now, if we desire to build the hexagons on this cube, we must define the degrees associated to the two new corners (U) = (A) ⊔ (E) and (Y) = (I) ⊓ (O), where ⊔ and ⊓ are, respectively a union and an intersection operation. If we consider as intersection the nilpotent triangular norm used to define the cube and its dual triangular conorm as union, we get a perfect fuzzification of the hexagon as explained in Section 4.3.

Also in this case, we can build a hexagon on the top and obtain the fuzzified version of the square product between the two relations R and S. In order to exactly match De Baets and Kerre definition, it is sufficient to take as conjunction of (a) and (A) the min operator.

Conclusion

For a long time, one has been interested in particular instantiations of the square and hexagon of opposition, and these structures have been found fruitful to lay bare all operators of interest in an exhaustive and structured way in these instantiations. In this paper, we have looked at the structures associated to the square, cube and hexagon for themselves, showing that they can be generated by a simple device based on the composition of a relation and a subset. The fact that relations and subsets are encountered everywhere makes the universality of these structures even clearer. Moreover, their study naturally suggests the question of generalizing these structures when the statements corresponding to the vertices are associated with degrees. We have shown how this extension enables us to retrieve the different ways of composing fuzzy relations. These graded structures apply to a variety of settings useful in knowledge representation, including modal logic [START_REF] Dubois | Knowledge-driven versus data-driven logics[END_REF], possibility theory [START_REF] Dubois | Gradual structures of oppositions[END_REF], rough set theory [START_REF] Ciucci | The structure of oppositions in rough set theory and formal concept analysis -Toward a new bridge between the two settings[END_REF], formal concept analysis [START_REF] Dubois | From Blanché's hexagonal organization of concepts to formal concept analysis and possibility theory[END_REF], abstract argumentation [START_REF] Amgoud | A formal concept view of abstract argumentation[END_REF], and analogical proportions [START_REF] Miclet | Analogical proportions and square of oppositions[END_REF]. It should contribute to bridge the gaps between these settings, and may induce new developments in each of them. These are topics for further research.

  (a) A and O are the negation of each other, as well as E and I; (b) A entails I, and E entails O; (c) A and E cannot be true together, but may be false together; (d) I and O cannot be false together, but may be true together.

Proposition 1

 1 In a square of opposition AIEO, the following holds: (a) The diagonal link between A and O, which represents the symmetrical relation of contradiction, is an exclusive or, namely ¬(A ≡ O), or if we prefer A ≡ ¬O. Similarly, E ≡ ¬I. (b) The vertical arrows represent implication relations A → I and E → O. (c) The link between A and E which represents the symmetrical relation of contrariety, corresponds to mutual exclusion, namely ¬A ∨ ¬E should hold. (d) The link between I and O which represents the symmetrical relation of subcontrariety, is a disjunction, namely I ∨ O holds.

Proposition 2

 2 Let AIEO be a square of opposition and (a)-(d) the conditions listed in Proposition 1. Then, (dep 1) (a)(b)⇒(c): ¬A ∨ ¬E is a consequence of A ≡ ¬O and E → O (or of E ≡ ¬I and A → I) in the square. But, ¬A ∨ ¬E and E → O (resp. A → I) are not enough for entailing A ≡ ¬O (resp. E ≡ ¬I). (dep 2) (a)(b)⇒(d): I ∨ O is a consequence of A ≡ ¬O and A → I (or of E ≡ ¬I and E → O). But, I ∨ O and A → I (resp. E → O) are not enough for entailing A ≡ ¬O (resp. E ≡ ¬I). (dep 3) (a)(c)⇒(b)(d) & (a)(d)⇒(b)(c): A ≡ ¬O, E ≡ ¬I, together with ¬A ∨ ¬E entail A → I, E → O and I ∨ O. Similarly, A ≡ ¬O, E ≡ ¬I, together with I ∨ O entail A → I, E → O and ¬A ∨ ¬E. Let us summarize the situation. One may consider that the involutive nature of negation expressed by (a) is crucial. Moreover, note that the formulas encoding (b), (c), and (d) together do not entail the formulas associated with (a). Besides, on the one hand conditions (a) and (b) entail conditions (c) and (d), while on the other hand conditions (a) and (c) entail conditions (b) and (d) (or in a dual manner, conditions (a) and (d) entail conditions (b) and (c)). This leaves us with three options for defining a formal square of opposition with independent conditions: i) either we can regard (a) and (b), or ii) (a) and (c), or iii) (a) and (d), as the basic requirements.

  ) = {x ∈ X | ∀s ∈ S, ¬(xRs)} = ) = {x ∈ X | ∀s ∈ S, ¬(xRs)} = {x ∈ X | xR ⊆ S} = s∈S Rs = s∈S Rs

Fig. 2

 2 Fig. 2 Square of opposition induced by a relation R and a subset S

R

  (S) = {x ∈ X | ∃s ∈ S, ¬(xRs)} = s∈S Rs and their complements R(S) = {x ∈ X | ∀s ∈ S, xRs} = ) = {x ∈ X | ∀s ∈ S, xRs} = {x ∈ X | S ⊆ xR} = s∈S Rs = s∈S Rs

Fig. 3 Fig. 4

 34 Fig. 3 Cube of opposition induced by a relation R and a subsetS

Fig. 5 Definition 2

 52 Fig. 5 Left side facet of the cube of opposition induced by a relation R and a subset S

Definition 3

 3 Given a square of opposition AEIO, a hexagon of opposition is made of the four square vertices plus the bottom vertex Y = def I ∧ O, and the top vertex U = def A ∨ E. The hexagon fully exhibits the logical relations inside a structure of opposition generated by the three mutually exclusive situations A, E, and Y (Y is the conjunction of the negation of A and the negation of E), where two vertices linked by a diagonal are contradictories, A and E entail U, while Y entails both I and O. Moreover it can be checked that I = A ∨ Y and O = E ∨ Y. Conversely, three mutually exclusive situations playing the roles of

Fig. 7

 7 Fig. 7 Hexagon induced by the left-hand side square

  [0,[START_REF] Amgoud | A formal concept view of abstract argumentation[END_REF] to each vertex A, E, O, and I of the square of opposition, namely α, ǫ, o, and ι respectively. Let n be an involutive negation operator (i.e., n(1) = 0; n(n(s)) = s), and * a symmetrical conjunction (i.e., s * t = t * s; 0 * s = 0; 1 * s = s). Let ⊕ denote the De Morgan dual of * , namely, s ⊕ t = n(n(s) * n(t)), ⇒ the strong implication, i.e., s ⇒ t = n(s * n(t)) = n(s) ⊕ t, associated with * . Note that s ⇒ t = 1 is equivalent to s * n(t) = 0. Definition 4 A graded square of opposition α, ǫ, o, ι is a square where each corner is associated to a degree in [0, 1] and constraints (a)-(d) are rendered as follows. (a) A and O are contradictories, i.e., the negation of each other, as well as E and I, can be encoded by α = n(o) and ǫ = n(ι) (b) A and I are sub-alterns, so A entails I. Also E and O are sub-alterns and E entails O. This translates into α ⇒ ι = 1 and ǫ ⇒ o = 1, i.e., α * n(ι) = 0 and ǫ * n(o) = 0; (c) A and E are contraries, hence they cannot be true together, but may be false together. It can be encoded by α * ǫ = 0 or equivalently n(α * ǫ) = 1; (d) I and O are sub-contraries, hence they cannot be false together, but may be true together. It can be encoded by n(ι) * n(o) = 0 or equivalently n(n(ι) * n(o)) = 1, i.e., ι ⊕ o = 1. It is straightforward to check that we still have for any involutive negation and any symmetrical conjunction that conditions (a) and (b) entail conditions (c) and (d) (i.e., dep 1 and dep 2 in Proposition 2), conditions (a) and (c) entail conditions (b) and (d), and conditions (a) and (d) entail conditions (b) and (c) (i.e., dep 3). Let us consider the various algebraic structures that can support the square of opposition. Kleene systems Taking min for the conjunction * , we get the constraints (a) α = n(o); ǫ = n(ι); (b) min(α, n(ι)) = 0 and min(ǫ, n(o)) = 0; (c) min(α, ǫ) = 0; (d) max(ι, o) = 1.

Proposition 4

 4 Any system ( * , n) with * a nilpotent triangular norm and n(x) = 1x that satisfies conditions (a) α = n(o), ǫ = n(ι), (b) α ≤ ι and ǫ ≤ o, (c) α * ǫ = 0, (d) n(ι) * n(o) = 0 satisfies the dependencies dep 1, dep 2, and dep 3.

Definition 5

 5 Given an involutive negation n, a symmetrical conjunction * , and using the strong implication associated with * , the constraints associated to the front facet and the back facet of the cube yield (a) α = n(o), ǫ = n(ι) and α ′ = n(o ′ ) and ǫ ′ = n(ι ′ ); (b) α * n(ι) = 0, ǫ * n(o) = 0 and α ′ * n(ι ′ ) = 0, ǫ ′ * n(o ′ ) = 0; (c) α * ǫ = 0 and α ′ * ǫ ′ = 0; (d) n(ι) * n(o) = 0 and n(ι ′ ) * n(o ′ ) = 0.

Proposition 9

 9 Let α, ǫ and γ be three numbers such that α + ǫ + γ = 1 (a fuzzy partition). If we consider the three other numbers ι = n(ǫ), o = n(α), ν = n(γ ), where n = 1 -(•) is the standard involutive negation and * the Łukasiewicz triangular norm, we obtain a hexagon of opposition. Proof It is clear that α * ǫ = α * γ = γ * ǫ = 0, since α + ǫ ≤ 1 and so on. Moreover, we have α→ ι = min(1, 1α + 1ǫ) = 1 (since α + ǫ ≤ 1); we also have α → ν = min(1, 1α + 1γ ) = 1, and γ → ι = min(1, 1γ + 1ǫ) = 1.It can be checked that all the other hexagonal constraints are satisfied as well.

(

  I) R • S = {(x, z)|∃y : xRy and ySz} = {(x, z)|xR ∩ Sz = ∅} (E) R • S = {(x, z)|∀y : xRy ⇒ ¬ySz} = {(x, z)|xR ⊆ Sz} (A) R • S = {(x, z)|∀y : xRy ⇒ ySz} = {(x, z)|xR ⊆ Sz} (O) R • S = {(x,z)|∃y : xRy and ¬ySz} = {(x, z)|xR ∩ Sz = ∅}

  (o) R • S = {(x, z)|∃y : ¬xRy and ySz} = {(x, z)|xR ∩ Sz = ∅} (a) R • S = {(x, z)|∀y : xRy ⇐ ySz} = {(x, z)|Sz ⊆ xR} (e) R • S = {(x, z)|∀y : ¬xRy ⇒ ySz} = {(x, z)|∀y : xRy or ySz} = {(x, z)|xR ∪ Sz = Y } (i) R • S = {(x, z)|∃y : ¬xRy and ¬ySz} = {(x, z)|xR ∩ Sz = ∅}

  y) → S(y, z), sup y∈Y (R(x, y) * S(y, z)) (A ′ ) min inf y∈Y R(x, y) → S(y, z), sup y∈Y R(x, y), sup y∈Y S(y, z)

(

  Eb) min inf y∈Y R(x, y) → 1 -S(y, z), sup y∈Y R(x, y), sup y∈Y 1 -S(y, z)(Eb ′ ) min inf y∈Y R(x, y) → 1 -S(y, z), sup y∈Y (R(x, y) * 1 -S(y, z))However, with these definitions we immediately lose the possibility to define a square of opposition since (I), the standard * -based composition of fuzzy relations, and (E) are not complement of each other. We can proceed similarly for the back square. So, (a) is the fuzzification of the superproduct and (o) its negation:(a) min inf y∈Y S(y, z) → R(x, y), sup y∈Y (R(x, y) * S(y, z)) (o) max sup y∈Y 1 -(S(y, z) → R(x, y)), inf y∈Y (1 -R(x, y)) ⊕ (1 -S(y, z))Then, (i) the standard composition of R and S, and (e) its negation:(i) sup y∈Y [1 -(R(x, y) ⊕ S(x, y))] (e) inf y∈Y [R(x, y) ⊕ S(x, y)]Thus, we have a fuzzy cube of opposition built by the corners (A), (I), (E), (O) in the front face and (a), (i), (e), (o) in the back face, by considering a nilpotent triangular norm as conjunction.

In the following, we use the weak form of the entailment. Indeed if we want to keep the strong form, Y entails O, for instance, would translate into ι * o * n(o) = 0, which would require that * is such that s * n(s) = 0, which would lead to take * as a nilpotent triangular norm.


C Caër

R Pesci

Local behavior of an AISI 304 stainless steel submitted to in situ biaxial loading in SEM

The microstructural response of a coarse grained AISI 304 stainless steel submitted to biaxial tensile loading was investigated using SEM and X-ray diffraction. The specimen geometry was designed to allow for biaxial stress state and incipient crack in the center of the active part under biaxial tensile loading. This complex loading was performed step by step by a micromachine fitting into a SEM chamber. At each loading step FSD pictures and EBSD measurements were carried out to study the microstructural evolution of the alloy, namely grain rotations and misorientations, stress-induced martensite formation and crack propagation. According to their initial orientation, grains are found to behave differently under loading. Approximately 60% of grains are shown to reorient to the [110] Z orientation under biaxial tensile loading, whereas the 40% left undergo high plastic deformation. EBSD and XRD measurements respectively performed under loading and on the post mortem specimen highlighted the formation of about 4% of martensite.

Introduction

The drive toward miniaturized systems in various engineering fields motivates the study of materials behavior close to their industrial scales. Moreover most of the phenomena occurring at the macroscopic scale result from mechanisms (phase transformation, cracking,..) happening at the microscopic scale.

Numerous experimental means and techniques have therefore been adapted from macroscopic to nanometric scale. This allows to study the influence of environmental constraints such as mechanical loading [START_REF] Roa | 8. (a) FSD picture of the crack in the studied area and (b), (c) and (d) crack reported on the EBSD mappings of respectively the initial state, the 4[END_REF][START_REF] Ma | Influences of tensile pre-strain and bending pre-deflection on bending and tensile behaviors of an extruded AZ31B magnesium alloy[END_REF], temperature [START_REF] Muñoz-Moreno | Analysis of crystallographic slip and grain boundary sliding in a Ti-45Al-2Nb-2Mn (at%)-0.8vol%TiB 2 alloy by high temperature in mechanical testing[END_REF][START_REF] Guo | Temperature-dependent mechanical properties of an austenitic-ferritic stainless steel studied by in situ loading in a scanning electron microscope (SEM)[END_REF], speed [START_REF] Vaz-Romero | The deterministic nature of the fracture location in the dynamic tensile testing of steel sheets[END_REF][START_REF] Taylor | Dynamic behavior of 304 stainless steel during high Z deformation[END_REF] or even irradiation [START_REF] Reichardt | In situ micro tensile testing He2+ ion irradiated and implanted single crystal nickel film[END_REF][START_REF] Dong | Microchemical and microstructural evolution of AISI 304 stainless steel irradiated in EBR-II at PWR-relevant dpa rates[END_REF] on the material behavior at a small scale [START_REF] Borkar | In-situ EBSD study of deformation behavior of Al-Si-Cu alloys during tensile testing[END_REF]. Numerous studies can be found in the literature regarding the microstructural evolution of materials under in situ uniaxial tensile tests [START_REF] Wang | Study of twinning/detwinning behaviors of Ti by interrupted in situ tensile tests[END_REF][START_REF] Boehlert | In situ analysis of the tensile and tensile-creep deformation mechanism in rolled AZ31[END_REF]. However, during their real service life, materials are usually loaded according to far more complex loading paths than uniaxial tension. Different mechanical loading paths available on macroscopic tests benches (bulge test [START_REF] Livescu | Biaxial deformation in high purity aluminum[END_REF], bending [START_REF] Mattei | Strain localization and damage mechanisms during bending of AA6016 sheet[END_REF], …) were therefore adapted to smaller scales.

The present work aims at studying microstructural features evolution under a specific complex mechanical loading, namely equi-biaxial tensile loading, using laboratory facilities. This kind of study has already been developed for Synchrotron [START_REF] Collins | A synchrotron X-ray diffraction of in situ biaxial deformation[END_REF][START_REF] Erinosho | Assessment of X-Ray diffraction and crystal plasticity lattice strain evolution under biaxial loading[END_REF] or neutron diffraction [START_REF] Marin | Diffraction measurements of elastic strains in stainless steel subjected to in situ biaxial loading[END_REF][START_REF] Van Petegem | In-situ neutron diffraction during biaxial deformation[END_REF]; however it is important to set up such a loading path in more ordinary equipment such as Scanning Electron Microscopes (SEM). This work involves an approach that combines Electron Backscattered Diffraction (EBSD) and X-ray diffraction (XRD).

The second section of this paper focuses on the presentation of experimental details, advanced techniques and materials used in this work. AISI 304 steel was used in this study as it is the most extensively used steel finding applications in the automotive and nuclear industries in particular. In situ experiments were performed using a miniature biaxial tensile test bench fitted in the SEM and under X-ray diffraction.

Although reference standards for uniaxial tension [START_REF]Metallic Materials -Tensile Testing -Part1: Method of Test at Room Temperature[END_REF] or bending [START_REF]Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature[END_REF] impose to use specific specimen geometries, the current standard for biaxial tensile tests [START_REF]Metallic Materials -Sheet and Strips -Biaxial Tensile Testing Method using a Cruciform Test Piece[END_REF] is less constraining by allowing the use of alternative specimen geometries [START_REF] Xiao | Biaxial tensile testing of cruciform slim superalloy at elevated temperatures[END_REF][START_REF] Kubo | Development of biaxial tensile test system for in-situ scanning electron microscope and electron backscatter diffraction analysis[END_REF]. The third part of this paper therefore presents a convenient specimen geometry determined from Finite Element Method (FEM) simulations to in situ validation in SEM. The newly designed sample geometry should allow for a biaxial stress state in the active part of the specimen and crack initiation in the center of said active part.

From these in situ biaxial tensile tests performed in SEM, the study of mechanical and microstructural features evolution under a complex loading path was performed. Microstructural evolution was first observed according to features such as grain reorientations and misorientations under loading. A particular focus was also made on the stress-induced martensitic transformation. Finally this study focuses on crack initiation and propagation.

⁎ Correspondence to: Arts et Métiers ParisTech, 4 rue Augustin Fresnel, 57070 Metz, France. E-mail address: celia.caer@gmail.com (C. Caër).

MARK

Materials and methods

Material processing and preparation

The investigated alloy is an austenitic AISI 304 stainless steel rolled in cylinders of 100 mm diameter; the chemical composition is given in Table 1. This material showed in the as-received state a yield strength σ y =325 MPa, a maximum strength σ m =1000 MPa and a Young mod- ulus E = 200GPa [START_REF] Rodríguez-Martínez | Experimental survey on the behavior of AISI 304 steel sheets subjected to perforation[END_REF]. The as-received material was examined using EBSD which showed a too fine-grained microstructure for the study to be carried out. The steel cylinder was therefore annealed in vacuum at 1200 °C for 1 h to increase the grain size up to an average of 300 µm.

Thanks to electro-discharge machining, slices were cut out of the steel cylinders and samples were machined from the slices.

Prior to any EBSD analysis, sample surfaces were carefully prepared. Specimens were first mechanically polished with decreasing SiC papers and diamond particle pastes. The final mechanical polishing was performed with a 1 µm diamond paste. The samples were finally electropolished in a solution of (C 2 H 4 O 4 + H 2 O) at 16 V for 1 min.

Biaxial tensile testing

Samples for macroscopic biaxial tensile tests were mounted on a servo-hydraulic multiaxial testing machine developed by INSTRON. The loading was chosen to follow a balanced biaxial tensile loading path (ε /ε =1 TD RD

) path up to failure. The experiments were conducted using a constant strain rate of 1.5*10 -2 s -1 . An infrared camera was placed in front of the specimen in the biaxial tensile bench in order to follow the temperature evolution and the associated stress concentrations during the whole loading.

In situ mechanical tests were performed thanks to a micromachine (model called "Proxima") developed by the company MICROMECHA. This machine, dedicated to mechanical testing in SEM and with XRD (Fig. 1), was fitted with a tensile biaxial set up for this study. In situ equi-biaxial tensile tests have been performed at low strain rates around 1.5*10 -3 s -1 . This experimental setup allows for:

-The quantification of the amount of martensite formed during loading when placing the post-mortem specimen on a XRD goniometer.

-The study of the microstructure evolution (texture, intragranular misorientations, plasticity, phase transformation…) during loading when mounting the micromachine into a SEM.

X-rays diffraction technique

XRD is a very efficient non-destructive technique which offers the opportunity to not only study crystallographic texture but also accurately quantify the volume fraction of phases and evaluate the level of residual stresses. XRD has been used in this study to measure the volume fraction of both austenite and martensite phases in the postmortem specimens following the ASTM E975-13 standard.

The volume fraction quantification of the two phases contained in AISI 304 steel has been performed with a PROTO iXRD goniometer fitted with a 1D detector by taking into account the intensity of the diffraction peaks of both austenite (A) and martensite (M); two peaks of each phase were retained as summed up in Table 2. Because of the two-phase microstructure of the steel, the sum of the volume fractions of austenite V A and martensite V M should be equal to 1.

The volume fraction of austenite is obtained according to (Eq. ( 1)):

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ∑ ∑ ∑ q I R p I R q I R V = 1 / / 1 / + 1 / j q Aj Aj i qp Mi Mi j q Aj Aj A =1 =1 =1 (1)
where I Mi and I Ai are respectively the diffraction peak intensities of martensite and austenite. R Mi and R Ai are coefficients depending on the Lorentz Polarization and the Debye-Waller factors, multiplicity of the considered {hkl} reflections and the crystal lattice of the phase [START_REF]Standard Practice for X-Ray Determination of Retained Austenite in Steel with Near Random Crystallographic Orientation[END_REF]. The volume fraction of both austenite and martensite have been measured using a collimator 1 mm in diameter and a chromium tube (λ=2.29 Å) which allows a penetration depth of about 10 µm. The uncertainty in the phase fraction measurement is about 0.3%.

Scanning Electron Microscopy

The device used in this study is a 7001 Field Electron Gun Scanning Electron Microscope from JEOL equipped with an Oxford EBSD CCD camera fitted with FSD detectors. EBSD data were post-treated with the software CHANNEL 5 from Oxford [START_REF] Prasad | Electron back scattered diffraction characterization of thermomechanical fatigue crack propagation of a near α titanium alloy Timetal 834[END_REF].

Given the coarse grain size, step size was set to 1 µm and the voltage to 15 kV. The eventual field of view obtained was 1.2*1.2 mm.

The biaxial tensile micromachine was set in the SEM and the loading was applied step by step until specimen failure. 15 steps were applied up to failure. SEM images, ESBD mappings and FSD images were performed at each loading step allowing for the microstructure characterization at each stop-off. An effort was made to keep the same area, namely the center of the specimen, under the field emission gun during the whole loading.

The EBSD technique was operated to study the evolution of [START_REF] Ktari | On the interaction between transformation induced plasticity and the austenitic stainless steel anisotropy (AISI 304) under shear loading path[END_REF]:

grain reorientations and misorientations, phase transformation as the software was required to identify the crystal lattice of austenite (FCC) and martensite (BCC).

Specimen geometry design for in situ loading in SEM

This section details the process of specimen geometry design for biaxial tensile experiments. This specimen geometry should allow for: biaxial stress state in the active part of the specimen maximum stress and therefore crack initiation in the center of the active part under equi-biaxial tensile loading.

Due to the cost of machining specimen at the microscopic scale, the geometry design of the specimen was first done at the macroscopic scale thanks to FEM simulations and experimental tests before being adapted to the SEM scale.

Numerous specimen geometries designed for biaxial tensile tests are found in the literature. Among those, some have been designed to obtain a homogeneous biaxial stress state in the gauge area either thanks to thickness reduction of the active part [START_REF] Makris | Shape optimization of a biaxially loaded cruciform specimen[END_REF][START_REF] Yu | Design of a cruciform biaxial tensile specimen for limit strain analysis by FEM[END_REF] or slit cuts into the arms [START_REF] Kuwabara | Measurement and analysis of differential work hardening in cold-rolled steel sheet under biaxial tension[END_REF][START_REF] Hanabusa | Numerical verification of a biaxial tensile tests method using a cruciform specimen[END_REF]. As the aim of the present study was to determine a macroscopic geometry that could also be adapted at the microscopic scale, the geometry with "slit cuts into the arms" geometry was considered as being too difficult to machine at the SEM scale. This study therefore focuses on reducing the volume of the gauge area to reach the aforementioned two conditions. Baptista et al. [START_REF] Baptista | Optimization of cruciform specimens for biaxial fatigue loading with direct multi search[END_REF] used cruciform specimen showing round thickness reduction on both sides of the active part. They obtained a maximum stress leading to damage at the center of the gauge area. In the present study, one of the requirements was to keep one face perfectly flat for the specific EBSD surface preparation and observation. Then thickness reduction could not be operated on both sides of the specimen. As a consequence the specimen was re-designed with thickness reduction of only one side of the active part.

Several specimen geometries were designed and biaxially loaded thanks to FEM simulations. The complete model has been imported into the V6.13-2 Abaqus (Dassault Systèmes) finite element calculation software. An isotropic elastic constitutive law has been adopted to model the material. A general static step has been considered in order to load the specimen, in the same configuration that in experiments in terms of boundary conditions. The specimen has been meshed with C3D10 quadratic tetrahedron continuous and isoparametric finite elements with a minimum size of 0.1 mm. This size allowed to have at least 3 elements in the thickness of the sample in order to well describe heterogeneous stress field. The geometry variations consisted in different shapes of the central thinned zone: round (cf. Fig. 2), square and clover were compared to specimens with no thickness reduction. Macroscopic specimens with these geometries were all tested; some of them showed some detrimental stress concentrations.

The most convenient geometry presented a round thickness reduction on one face of the specimen (32% of the initial thickness remaining) as shown on Fig. 3a. This specimen geometry was then machined to be tested on a macroscopic biaxial tensile test bench following an equi-biaxial loading path. During loading, the infrared camera placed in front of the specimen showed an increase in temperature corresponding to stress concentration at the center of the gauge area. Just before fracture, the maximum stress was localized in the very center of the active part, leading to crack initiation at this point (cf. Fig. 4a). The specimen geometry presented in Fig. 3a suited design and testing requirements well, it was therefore adapted to the lower scale.

The final geometry of the in situ specimen was adapted from the macroscopic specimen geometry by a proportional reduction of the active part and a re-design of the arms in order to fit the micromachine clamping system (Fig. 3b). The active part is a 3.6 mm wide and 1 mm thick square showing a round thickness reduction in the center (0.32 mm remaining). Prior to the micromachine being set up within the SEM, a pre-test was performed in order to validate the specimen geometry.

The test was performed ex situ under an optical microscope. An equi-biaxial tensile loading was applied to the specimen until failure. This experiment was led under the same loading conditions as the in situ tests and resulted in a crack in the very center of the active part of the specimen as presented in Fig. 4b. The geometry was therefore validated for in situ tensile biaxial tests.

Results

Force-displacement curve

The resulting force-displacement curve was used to determine the different loading steps applied in the following in situ biaxial tensile loading, as shown in Fig. 5. The curve representing engineering stress as a function of strain is not presented here as the thickness reduction makes the force/section relation no more accurate. The loading performed in the SEM was therefore operated under displacement control and stopped off every 200 µm. Five of these strain level interruptions are investigated in this study, namely: the initial state, the fourth load (end of the elastic behavior), the ninth load (approximately at half of the strain hardening), the fourteenth load (last load before fracture) and the fifteenth load (appearance of the first crack).

At each loading stop FSD images and EBSD mappings were performed, informing on grain reorientations and misorientations, martensite formation and crack propagation.

Microstructural evolution

In this section, focus is made on the evolution of microstructure before failure on the first four specific loading steps. For each loading step, FSD images and data extracted by EBSD (EBSD maps and poles figures) are considered, as shown on Fig. 6. FSD images taken at different loading steps show that grain deformation induces a relief formation at the free surface. These images also show that grain with twins due to annealing are present at the initial state, but no more twin appears during loading. Fig. 6 indicates the evolution of crystallographic texture obtained thanks to EBSD analysis for the initial state, the 4th, the 9th and the 14th loading steps. At the initial state, the specimen does not show specific texture despite the presence of coarse grains and intragranular misorientations are low. According to Fig. 6, at the end of the loading, the material shows a [101] Z texture with a maximum intensity factor of 4. Two main trends of grain behavior may be distinguished upon loading.

At the fourteenth load, grains show either a [101] Z orientation or important intragranular misorientations. Considering the grain numbering (Fig. 6a), grains 1-6 present a [101] Z orientation at the final loading steps. These grains showed an initial orientation close either to the [101] or the [111] Z direction.

Grains 7, 8 and 9 which show a Z orientation differing from [101] or [111] at the initial state do not undergo a global rotation but intragranular misorientations that strengthen with the loading. Quantitatively, the amount of grains reorienting to the [101] Z orientation reaches 60%, whereas approximately 40% undergo high plastic deformation under biaxial tension.

A gradual appearance of parallel bands on the grains surface is also observed on FSD images. Upon loading, these bands get more and more numerous, thicken and appear along different directions in a same grain. They may be due either to slip systems and/or to martensite variants activation [START_REF] Kermanpur | A microstructural investigation on deformation mechanisms of Fe-18Cr-12Mn-0,05C metastable austenitic steels containing different amounts of nitrogen[END_REF].

Martensite formation

As martensitic transformation was expected to be induced upon loading [START_REF] Haušild | Characterization of strain-induced martensitic transformation in a metastable austenitic stainless steel[END_REF], EBSD data was collected at each loading step for both austenite and martensite. The two steel phases present different crystal lattices: FCC for austenite and BCC for martensite (actually, martensite presents a tetragonal lattice but the ratio between c and a lattice parameters is so negligible that martensite is assimilated to BCC).

Resulting phase mappings are showed on Fig. 7 represents phase distributions and grain boundaries at the initial state and at the fourteenth load. At the initial state, all grains show a uniform yellow color, meaning a fully austenitic state. Martensite formed upon loading appears in blue on the fourteenth load (cf. Fig. 7b).

At the initial state, EBSD mappings show that the specimen is fully austenitic, no evidence of martensite is found. Then, at the fourth load, the first evidence of martensite formation is detected and measured up to 0.2% ± 0.05%. The martensite ratio increases with the applied load: it reaches 0.4% ± 0.1% at the ninth load to end at 3.8% ± 0.4% at the last load before failure. In order to validate these results collected thanks to the EBSD set up, XRD phase quantification was performed on the specimen, after failure. These measurements, carried out according to the details presented in section II, showed that the post mortem specimen contained 4% of martensite; the uncertainty on the volume fraction measurement being about 0.3%. The difference of 0.2% in martensite between EBSD and XRD measures may be explained first by the different techniques used and their uncertainties, but also by the fact that EBSD results were obtained before failure and XRD measurements were made on the post mortem specimen that had undergone a slightly higher load.

Crack propagation

The biaxial tensile loading was applied to the specimen until the fifteenth load when the first crack appeared. An incipient crack appeared in the center of the active part and propagated along a diagonal from fillet to fillet as expected given the specimen geometry. Crack propagation occurred across the area of interest allowing the study of fracture governing mechanisms under equi-biaxial tension.

It is known that under high tensile stress, crack propagation results from two competing damage mechanisms, namely intergranular cracks (propagation along grain boundaries) and transgranular cracks (i.e. propagation through the grains) [START_REF] Taheri Mousavi | Dynamic crack propagation in a heterogeneous ceramic microstructure, insights from a cohesive model[END_REF][START_REF] Liu | Mechanics and Mechanisms of Fracture: An Introduction[END_REF]. The type of crack propagation resulting from high biaxial stresses is here studied.

To begin with, the crack propagation path was identified on the FSD image obtained just after failure (cf. Fig. 8a) and drawn on the inverse pole figure map obtained by EBSD at the initial state (cf. Fig. 8b), the fourth load (cf. Fig. 8c) and the fourteenth load (cf. Fig. 8d). The crack appears to mostly propagate along grain boundaries (Fig. 8b andc), in the upper part of the studied area where the crack propagates through high strained areas, in 'grain 1' (cf. Fig. 6a) encircled on Fig. 8d where the crack propagates through the grain.

Discussion

The EBSD indexation rate decreases upon loading from 100% at the first loading steps to 85% at the fourteenth load (cf. Figs. 6 and7 by the appearance of white areas corresponding to non-indexed areas). This is due to two major effects. On the one hand, a masking effect appears with the surface relief: some parts of grains get off the surface and mask parts of other grains. This effect is magnified by the 70°tilt necessary for EBSD analysis. On the other hand, EBSD indexation rate also decreases due to high strains undergone by the material and its crystal lattice. Indeed, at the end of the loading, the crystal lattice of some grains is so highly deformed that the Kikuchi bands corresponding to the diffracted signal are blurred. Then the software can no longer index the resulting patterns in these regions. However, the 15% drop of the indexation rate is weak enough for the fourteenth load data to be handled.

Microstructural response to tensile biaxial loading

Results described in Sections 4.1 and 4.2 show that the biaxial tensile loading acts on microstructure at different scales. At the grain scale, loading induces grain reorientation according to the initial grain orientation. Grains with an initial orientation, close to [111] and [101] rather reorient to or close to the [101] Z direction, whereas grains showing other initial orientations rather display plastic deformation. At the intragranular level, EBSD analysis shows that non-reoriented grains develop large deformations noticeable thanks to high intragranular misorientations.

Evidence of martensite formation upon loading has been made by both EBSD and post mortem XRD. According to results shown in Sections 4.2 and 4.3, it seems clear that martensite is localized in highly deformed areas. So either martensite formation leads to important lattice deformations in its neighborhood, or high intragranular misorientations induce the formation of martensite. No evidence of martensite formation is found in grains reorienting to a [101] Z orientation.

Some slip bands are present at the initial state due to mechanical polishing. Other slip bands appear during loading whether in reorienting and in grains presenting high misorientations. Some grains such as grain 7 even show cross slip due to the activation of two different slip systems, maybe due to the initial orientation and the influence of the surrounding grains.

Thus the material accommodates tensile biaxial loading by both intergranular and intragranular mechanisms: grain rotations and misorientations coupled to plasticity and martensitic transformation.

It has been shown previously that initial grain orientation has a great influence on its behavior upon biaxial tensile loading. The same statement was observed under uniaxial tension by Chakrabarty et al. [START_REF] Chakrabarty | Crystallographic orientation and boundary effects on misorientations development in austenitic stainless steel[END_REF] and modeled by Raabe et al. [START_REF] Raabe | Theory of orientation gradients in plastically strained crystals[END_REF]. Both authors observed a strong influence of the initial crystallography on misorientations development.

Another noteworthy result here is that by performing a uniaxial tensile loading path, the grain reorientation would have been expected in the [100] Z orientation. Under the biaxial tensile loading path here presented, the grain reorientation happens rather toward the [101] Z orientation showing that a change in the loading path induces a change in the microstructure evolution trend.

The instability of austenite in fully austenitic AISI 304 stainless steels makes it possible to form strain-induced martensite. This unstable austenite is sensitive to many factors such as temperature, loading path and stress triaxiality whereas grain size was shown not to affect the amount of deformation-induced martensite [START_REF] Mastuoka | Effect of grain size on thermal and mechanical stability of austenite in metastable austenitic stainless steel[END_REF]. Rodríguez-Martínez et al. [START_REF] Kisko | The influence of grain size on the strain-induced martensite formation in tensile straining of an austenitic 15Cr-9Mn-Ni-Cu stainless steel[END_REF] uniaxially loaded a material showing the same chemical composition and finer grains than the one used in this study. In the case of this particular loading, the fine-grain 304 stainless steel produced a volume fraction of martensite of approximately 70% in the necking area. In the present study the maximum volume fraction of martensite formed under biaxial tension reaches 4% and has been measured by XRD on the crack boundary. It appears therefore that loading path has a great influence on martensitic transformation ratio. Thus it seems that biaxial tension is less favorable to martensitic transformation than uniaxial tension. However one has to be aware that comparing these two loading paths at equal deformation is not so simple due to the thickness reduction of the biaxial specimen.

Crack propagation upon biaxial tensile loading

As expected for the ductile fracture of stainless steel at room temperature, plastic deformation occurred by transgranular slip during loading until it culminated in the formation and propagation of a crack along strain localization regions. A transgranular crack propagation mode is then expected due to the ductile fracture. However, according to Fig. 8b andc, the crack appears to mostly propagate along grain boundaries. This unexpected crack propagation mode may occur under ductile fracture of a heat treated material. In the present study, the annealing operated to increase the grain size may have caused segregation of certain impurities to the grain boundary [START_REF] Liu | Mechanics and Mechanisms of Fracture: An Introduction[END_REF], thereby embrittling the material at the grain boundary level.

It has been shown that transgranular cracks initiate near grain boundaries, when intergranular propagation becomes barely possible due to the local geometrical arrangement [START_REF] Musienko | Simulation of inter-and transgranular crack propagation in polycrystalline aggregates due to stress corrosion cracking[END_REF]. Concerning the crack propagation in high strained areas, the notions of grain and grain boundary no longer make sense at the very last loading steps (cf. Fig. 8d). Thus the reporting of crack propagation on the first EBSD mappings in the upper area undergoing very high strains and intragranular misorientations does not mean that the crack propagated through the grains, but through a deeply deformed lattice. This shows that the energy needed to propagate the crack through these highly strained areas is lower than along embrittled grain boundaries.

Focus is now made on the evolution of 'grain 1' orientation upon loading. At the initial state, except two localized twins, this grain shows a homogeneous {2 3 5} 2 3 0 orientation while at the fourth load, the initial grain splits into two sub-grains showing respectively {2 1 3} 8 13 1 and {2 0 3} 3 6 2 grain orientation. Finally, the grain orientation returns to homogeneity at the fourteenth load with the {1 0 1} 1 2 1 orientation. Fig. 8c shows the crack propagates in a transgranular mode along the boundary between the two sub-grains of 'grain 1' appearing at the fourth load. Thus even if 'grain 1' appears to be completely homogeneous on the EBSD maps just before failure (cf. Fig. 6d), the crack propagates through some intragranular discontinuity undetected by the EBSD framework.

Conclusion

In this work the microstructural evolution of coarse grain AISI 304 stainless steel sheets subjected to biaxial tension has been experimentally studied. Conclusions of this study can be summed up as follows:

-A specimen geometry allowing for biaxial stress state and crack initiation in the center of the active part was designed at the macroscopic scale. Once the geometry was validated on a test bench, it was adapted to the microscopic scale in order to fit the micromachine for in situ biaxial tensile testing.

-At the grain level, it was shown that initial grain orientation has a strong influence on grain behavior upon loading. Depending on their initial orientation, some grains underwent high plastic deformation whereas other grains gradually rotated to a [101] Z orientation. Quantitatively, 60% of grains rotated to the [101] Z orientation, whereas 40% of the grains deformed plastically. -Coupled EBSD and XRD measurements of the austenite volume fraction showed that large deformations were accommodated by both phase transformation and plasticity. About 4% of martensite was formed in the active part under biaxial tension. -Crack propagation was shown to mostly occur at grain boundaries in an intergranular cracking mode. Except for high strained areas where notions of "grain" and "grain boundary" are difficult to apply.

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Fig. 2 .

 2 Fig. 2. FEM simulation of von Mises stress distribution on the rear side of a sample presenting a round thickness reduction and submitted to a proportional biaxial tensile loading.

Fig. 3 .

 3 Fig. 3. (a) Geometry of the tensile biaxial specimen with circle thickness reduction (b) geometry of the SEM scaled specimen (mm).

Fig. 4 .

 4 Fig. 4. a) IR image of the macroscopic cracked specimen under biaxial tensile loading, b) cracked microscopic specimen under tensile biaxial loading observed by optical microscopy.

Fig. 5 .

 5 Fig. 5. Load-displacement plot recorded during the in situ biaxial tensile test operated in the SEM. Every plot point represents a loading stop off; stars indicate specific loading steps investigated in this study.

Fig. 6 .

 6 Fig. 6. On the left: EBSD mappings, in the middle: FSD pictures and on the right: inverse pole figure of the center of the active part at (a) the initial state, (b) the fourth load , (c) the ninth load and (d) the fourteenth load.

  except:

Fig. 7 .

 7 Fig. 7. Mappings of the phase distribution obtained by EBSD (a) at the initial state and (b) at the fourteenth load, just before failure. In yellow: austenite, in blue: martensite and in black: grain boundaries.

Table 1

 1 Chemical composition of the AISI 304 stainless steel (% weight).

	C	M n	C r	N i	M o	C u	S i	N b
	0.006	1.54	18.47	8.3	0.3	0.37	0.48	0.027

Fig. 1. Micromachine from MICROMECHA company (Proxima model) mounted in a SEM and fitted with the tensile biaxial setting.

Table 2

 2 Planes analyzed and corresponding 2θ angles for both austenite and martensite.

	Phase	Austenite		Martensite
	Plane analyzed	{200]	{220}	{200}	{211}
	2θ angle (deg)	79	128.7	107.1	154.1