Valérie-Inés De

La Ville

Valérie Tartas

Developing as Consumers by

Aims  To introduce three perspectives used by marketing researchers and practitioners to explain how children develop as consumers : o a classic developmental-stage based perspective which focuses on the progressive acquisition of economic knowledge by children o a widespread process-centered model which aims at explaining how children interact with socialization agents to acquire economic abilities o a more recent framework drawing on cultural psychological theory that suggests that children are immersed in the realm of mass consumption culture  To discuss the cornerstone elements of these three models and the contrasting explanations they propose about the learning processes through which children develop as consumers  To delineate new research avenues in order to account for the creative capacities of children when they participate in joint consumption activities

Table 1: Stages in Children's Development as Consumers Characteristics Perceptual stage 3-7 years Analytical stage 7-11 years Reflective stage 11-16 years Knowledge structures Orientation

Introduction

Scientific articles aimed at linking the realms of childhood and economy appeared in the early 1950's and dealt mainly with the role children play when they take part in family consumption practices [START_REF] Strauss | The development and transformation of monetary meanings in the child[END_REF]. Thus consumer socialisation is a relatively recent research area as the concept itself was defined in 1974 for the first time by S. Ward: "The processes by which young people acquire skills, knowledge, and attitudes relevant to their functioning as consumers in the marketplace" (Ward 1974: 2). This broad definition calls for an interdisciplinary perspective, including social and cultural backgrounds, language skills, psychological development, emotional and symbolic dimensions, etc. [START_REF] Piaget | Children as Consumers: the need for multi-theoretical perspectives[END_REF]. Thus, this chapter aims to clarify the contributions of three established perspectives to the understanding of the development of children's abilities to behave as consumers. Moreover, in the process of growing up, children are involved in social relationships that go far beyond their direct or indirect participation in mere buying behaviours [START_REF] Oestergaard | Shifting perspectives in consumer research: from buyer behaviour to consumption studies[END_REF]. This means that through consumption activities, children learn a lot about contemporary society and the diverse social rules that prevail according to different social situations…

I -Developing as Consumers… by climbing a succession of cognitive stages in the use of economic notions

A first batch of marketing research draws on the seminal work by the Swiss psychologist Jean [START_REF] Piaget | Children as Consumers: the need for multi-theoretical perspectives[END_REF] and applies it to the analysis of the development of economic thinking by children. This research perspective has been driven by the question of "what" does the child know, the kind of economic knowledge the child is able to use when he or she is confronted with practical problems linked to consumption activities. Following Piagetian terms, development is regarded as the adaptation to the outside world, i.e. the construction by the individual of increasingly elaborated cognitive mechanisms to improve his or her control over the surrounding world. This developmental process unfolds in a natural way: by individually acting on external objects, the child gradually builds up skills related to his environment that will in turn impose its own structures on him or her. Thus, growing and learning imply passing through various stages of development which enable the child to control concrete logical operations, and later on, formal logical operations during adolescence. Development gives access to the increasingly complex logical designs that characterise adult thought which is regarded as the ultimate aim of child developmental processes.

A three-stage cognitive model of consumer development

The American researcher Deborah Roedder-John (1999) has documented 25 years of accumulated international research on children in relation to their role as consumers. She incorporated the findings into a general conceptual framework that conceives consumer socialisation as progressing through a series of three sequential stages capturing major cognitive shifts from preschool to adolescence: the perceptual stage (3 to 7 years), the analytical stage (7 to 11 years) and the learned stage (11 to 16 years) (see Table 1). These stages, essentially defined in terms of age groups, include various dimensions that characterise children's knowledge, such as cognitive and social skills, and motives during childhood and adolescence, and are also related to children's growing sophistication with regard to products, brands, advertising campaigns, shopping, pricing, decision-making strategies, and influential strategies. Using Piaget's theory, Roedder-John's framework seeks to identify the main shifts in children's understanding of basic economic concepts. It depicts young children's skills as being limited and only based on the perceptual skills which are not enough to understand implicit messages in advertising for example. Thus from 3 to 7 years old, children show an egocentric orientation, as they are not able to differentiate points of view, for example the persuasive perspective held by the advertiser and the need for information felt by the consumer. Moreover children's ability to deal with information is limited and prevents them from making decisions based on a plurality of dimensions, as adults do. When they gradually enter the analytical stage, this model provides descriptive information about the stages children go through, starting from fragmentary and imprecise ideas to a more logical understanding. In the analytical stage, children are becoming aware of the complexity of the market because their way of reasoning becomes more abstract and is no longer only driven by their own perceptions. The reflective stage allows pre-adolescents and adolescents to develop a critical thinking about the marketplace and how it functions. Based on this stage-oriented framework of child development, Roedder-John presented five areas of research that she considers are the effects of the process of consumer socialisation, as they imply an acquisition of various knowledge, skills and values, i.e. advertising and persuasion knowledge, transaction knowledge, and decision-making skills and abilities, negotiation skills, values and motives of consumption.

Perspective

Limits of this framework through a selection of three key critical comments

As a managerial transposition of the cognitive developmental theory of Piaget, this framework of consumer socialisation was mainly addressed to marketing managers with a two-fold aim: firstly, to describe the development of consumer socialisation through three cognitive stages and secondly, to offer a 'toolkit' to help marketing practitioners improve their targeting of children and adolescents. Roedder-John acknowledged that she focused on research only seen from a managerial and marketing perspective, and that she examined selected consumer research published in marketing and communication journals, which lead her to exclude other research areas dealing with child development. This might explain why her theoretical framework overlooked three important points:

 Cognitive development is driven by biological or internal factors:

To go beyond mere physical objects that were the main focus in Piagetian's studies, she used Selman's theory (1980) which integrates the social dimensions of child development and documents the changes in children's way of thinking about the social world… Although it takes into account children's capacity to consider anothers' point of view and to behave in accordance with a conventional system of social roles, this perspective ends in a very limited understanding of child consumer socialisation as it conceives socialisation itself as a fundamentally solitary cognitive construction by an individual. From a Piagetian perspective, the socialisation process of the child is seen only from the point of view of the endogenous factors that lead the child to pass from self-centredness to other-centeredness. In such a perspective, only internal processes may explain the development of economic thinking and the transitions between the different stages.

 The use of age as a marker of cognitive skills: by placing 'age' -instead of the internal structure of thought -as the cornerstone of her framework, Roedder-John diverted Piagetian theory from its own foundations. She conceived development only as a difference of degree in the understanding and use of economic knowledge, and not as a qualitative difference in the nature of reasoning involved in economic activities and market situations. Whereas, in Piagetian terms, the idea of stages is much more complex and relates to the internal cognitive mechanisms of assimilation, accommodation and equilibration. Jean Piaget himself stressed that what should be retained from his model is the succession of stages and not the age ranges because they could vary greatly according to cultural and social contexts (Bringuier 1977: 57). However, some researchers do criticise stage-oriented theories of child development as, in the context of a globalising marketplace fostering the internationalisation of mass media and consumer culture, the everyday life of contemporary children is extremely different from that of former times (Gunter a,d Furnham, 1998).

 The exclusive focus on logical reasoning: by considering children as 'economic problem solvers' trying to succeed in an adult economic world [START_REF] Duveen | Chidlren's savings: a study of development of economic behaviour[END_REF], managerial approaches promote a specific kind of knowledge and reasoning -logical -while overlooking a major mode of human thought, the narrative one. As [START_REF] Bruner | Acts of meaning[END_REF] pointed out, two major modes of thinking characterise human cognition: the 'syntagmatic' or 'narrative' mode and the 'paradigmatic' or 'logical' mode. In the first, events are connected to one another through a narrative frame allowing the creation of stories, whereas in a paradigmatic mode, reasoning faculties pursue logical truth. If the child develops his or her narrative capacity very early in everyday mundane contexts, the paradigmatic thought -or logic-scientific -only emerges after systematic education, for which the institutional dimension has received little consideration within the Piagetian theory (Cram and Ng, 1999: 300).

In short, we can conclude that the theory of cognitive development has played a dominant role in management academic circles as well as in marketing practice [START_REF] Siegel | The great tween buying machine -Capturing your share of the multibillion dollar tween market[END_REF]. Nevertheless, critics have pointed out that socialisation should be defined as a function of environmental influences applied to the person rather than as a purely individual cognitive process.

II -Developing as Consumers… by acquiring social skills and knowledge through interaction with socialisation agents

A second course of study endeavours to overcome certain limitations of the cognitive-driven approaches by incorporating the social aspects of economic development, using various social variables and the social education model elaborated by [START_REF] Bandura | Social learning theory[END_REF]. This process-oriented approach to consumer socialisation has gained ground in order to find out "how" the child acquires his or her economic knowledge.

-Child economic socialisation as a synthesis at the crossroads of three fields

Moschis and his colleagues [START_REF] Moschis | Consumer socialization: a theoretical and empirical analysis[END_REF][START_REF] Moschis | Decision making among the young: a socialization perspective[END_REF]) developed an alternative theory of child consumer socialisation that offered a more accurate definition of consumer socialisation. From this perspective, Lassare and Roland-Lévy's conceptual framework aims at linking four variables that derive from general socialisation theories: 1) the socialisation agents (peers, family, school, mass media); 2) the learning process (in particular imitation or vicarious learning); 3) the social structure variables (age, sex, social class); 4) the learning content [START_REF] Lassare | Understanding children's economic socialization[END_REF]. It represented a clear invitation to explore how the socialisation process actually works in everyday and casual social activities as well as in more formal social contexts.

Figure 2: Consumer Socialisation Process

(source: adapted from [START_REF] Moschis | Consumer socialization: a theoretical and empirical analysis[END_REF] This theoretical framework is an attempt to produce a consistent synthesis of three different perspectives. These three domains were considered inter-related, their combination allowing a broader understanding of economic socialisation processes.

First of all, the Piagetian theory revisited by Berti and Bombi (1989) which identified that children are active role-takers in the economic domain. Drawing on these studies, the authors defined economic socialisation as an on-going process by which the child assimilates knowledge and reasoning about the economic world and consumption practices. The emphasis was put on children's methods of interpreting the marketplace as well as on the specificity of the cognitive operations needed to understand economic notions.

Secondly, the social role model theory, which suggests that a child is supposed to learn how to play different roles in society, such as pupil, sibling, grandson, as well as… consumer. As far as the latter is concerned, the economic education is achieved in practice owing to four main sources of information: parents and family habits, peers, advertising and the products themselves. The inclusion of social roles emphasised the importance of social environment on children's cognitive elaborations. It considered the influences of parents and peers as well as the effect of gender as the main factors that help children become active economic agents -able to choose a product, to learn how to buy it and to understand how the marketplace is governed.

Socialisation agents

Parents, peers, school, media

Learning processes

Imitation, Reinforcement

Finally, [START_REF] Bandura | Social learning theory[END_REF] theory of social learning which suggests that two central processesobservation and imitation (mimicry of other consumers: parents, peers, brothers and sisters, etc.) -can explain both real consumer behaviour (such as comparing prices and products) and more complex and abstract notions (such as purchasing power and taxes). The role of social environment is explained through two main independent variables: on the one hand, the social environment in which the child is involved, and on the other, the nature of the reinforcement or feed-back provided by the environment.

The main contribution of this perspective is to have linked cognition to social interactions and to have exceeded the limitations of a strictly individualistic cognitive developmental approach. The authors focused on the way the child is integrated into society by the appropriation of the roles enacted by environmental models in family circles, peer groups and through mass media.

Limits of this framework through a selection of three key critical comments

Process-oriented theoretical frameworks about child economic socialisation perceive social environment through two key variables: the child's social environment and the feed-back and product reinforcement provoked by this same environment. Thus, this process approach tries to consider the individual and the cognitive operations involved in economic situations as well as the social contexts in which the child develops. However, in spite of its ambitious attempt, this framework leaves three important questions unanswered.

 A limited scope of exploration that ignores young children: this theoretical perspective allowed a recombination of several research results obtained through different methodologies: surveys through semi-structured interviews, face-to-face interviews in focus groups, observations and questionnaires. Nevertheless, the first shortcoming comes from the empirical studies that are mainly focused on pre-adolescents, adolescents, young adults and adults, neglecting young children such as preschool and early school children [START_REF] Lachance | College student's competence: identifying the socialization sources[END_REF]. As a result, this process perspective only covers a limited explanatory scope as it favours pre-adolescents and adolescents.

 Vicarious learning cannot account for children's creative capacities: the 'modelling exposure' perspective considers that the fact of being confronted by parental economic behaviour is sufficient to trigger the development of the child's economic skills. But then a crucial question remains unanswered: how can we explain the emergence of new behaviours in children, adolescents, or adults? Observation and imitation impede the explanation for the emergence of creative behaviour which does not just come from behaviours already developed within family circles or peer groups.

In psychology, the modelling approach has been criticised: imitation of adults or more experienced children is neither a necessary nor a sufficient condition to explain children's cognitive progressions [START_REF] Doise | Le développement social de l'intelligence[END_REF].

 Interaction does not constitute the key unit of analysis: although the process models highlight the role of others -individuals enacting their institutional roles -as a central explanatory factor, from a methodological standpoint, social interactions themselves do not appear as being the central focus of empirical analysis. In fact the individual and social issues are seen as two separate entities which certainly need to be re-entangled, but in the end, the individual remains the cornerstone of the analysis and the theoretical elaboration. Social influence approaches maintain a focus on the individual as the basic unit of analysis and examine the influence of 'outside' social forces. In the social influence perspective, since individual competence is traditionally seen as separate from environmental circumstances, individuals are considered to possess knowledge prior to being involved in a social interaction, then to have a social experience allowing consideration of other possible courses of action, and finally to absorb them so they become a part of their personal repertoire.

As [START_REF] Cram | Consumer Socialization[END_REF] noted, "both psychological and marketing approaches (social learning theory) have inherent faults in terms of their ability to provide a full picture of children's consumer socialisation. These faults include a lack of understanding of children as active agents within a cultural context which, in turn, has resulted in a dearth of cross-cultural research and the widespread acceptance of the white, middle-class nuclear family as the "normal" context within which children are socialised". They develop an approach called 'scaffolding' based on [START_REF] Bruner | Introduction[END_REF] 1) definition of socialisation as a process through which children acquire a "framework for interpreting experience and learning how to negotiate meaning in a manner congruent with the requirements of culture". Their 'scaffolding' approach ties three consumption-related concepts -'ownership', 'money' as a medium of exchange, and 'price' -which provide children with a framework for integrating an increasing variety of consumption experiences. The authors discussed how concepts validate and support children's consumer socialisation, leading to a wider perspective of consumer socialisation.

III -Developing as Consumers… by transforming children's participation in joint consumption activities

A more recent approach draws on cultural psychological theory and considers that children are immersed in the realm of mass consumption culture. From a theoretical standpoint, the emphasis is on examining how children actually participate in socio-cultural activities and especially how they take part in consumption activities. To understand how children learn and develop through participation in sociocultural activities, it is necessary to grant that cultural meaning is more than the mere addition of changes in individual thinking that result from social interaction and discussion with other participants (Rogoff 1998: 686). In this perspective, developmental transitions are approached through the transformation of children's roles in various socio-cultural activities related to consumption.

Consumption as a mediated and mediating joint activity

Within this framework, the child is clearly integrated into a social body with which he or she interacts and from which he or she receives help and reassurance to dare participate in mundane social life. Consequently, the relevant unit of analysis is certainly not the isolated child confronted with a problem to be dealt with, but rather the joint activity developed within an interaction 'child -adult', or 'childmore experienced child' or a 'small group' confronted with a practical problem -purchase decision, defining a socially acceptable use of the object… By engaging in a joint activity with a more experienced partner, the child discovers the cultural meaning of objects.

Vygotsky used the concept of 'psychological' or 'semiotic' tool to account for the relationship between the individual and the external world [START_REF] Wertsch | Vygotsky and the social formation of mind[END_REF][START_REF] Rogoff | Cognition as a collaborative process[END_REF]. Child consumption constitutes a social activity mediated by various cultural tools such as the language, the social standards called upon, the rhetoric invoked, and the purchased objects or products themselves. Through mundane activities, the child gradually integrates a conventional language peculiar to consumption -such as concepts of brands, prices, quality, comparisons between products -and a set of social standards relating to consumption in a given cultural context -such as mastered references, search for identity, membership within a group, and feeling of belonging to a social class [START_REF] Gronow | Ordinary Consumption[END_REF]. Thus, studying the psychological tools or the systems of signs used by the child during his or her development as an economic actor, enables a precise analysis of how children understand and practice consumption. To obtain what he or she wishes, the child must tell an appropriate story and know how to present its actions and objectives in order to make them legitimate within specific social groups: family circles or peer groups. As the narrative capacities of the child increase in the process of economic socialisation, researchers can analyze the evolution of children's accounts of their consumption practices. Towards 3 to 4 years, children can be regarded as 'experts' in the use of various narrative forms, knowing how to handle different rhetorical registers to evoke empathy in their close relations, which explains how children influence family purchases.

This leads us to consider that children develop as economic actors within a complex cultural system that combines several interdependent dimensions (see Figure 3). In the joint consumption activity, three elements are interwoven:

-The child is never alone but is always guided by others -adult caregivers and more experienced children -to question the practices of consumption and to construct socially shared meanings.

-The term 'cultural object' covers any socially and historically created product (not only the material and physical characteristics of the object but also its symbolic and semiotic dimension).

-The cultural object on which attention is focused is a socially and historically created product that includes in its design a technical, social and commercial background. This institutional background strongly directs its potential uses. The meaning of children's consumption is framed both by the normative demands of the family's consumption style and the normative requirements of peers' consumption styles. The child has to learn to deal with the conflictive requirements held by the different social groups to which he or she belongs [START_REF] Page | The impact of consumer environments on consumption patterns of children from disparate socioeconomic backgrounds[END_REF]. Moreover, beyond these face-to-face relationships, children's consumption practices are also enabled and constrained by wider institutional systems including:

JOINT CONSUMPTION ACTIVITY CO-CONSTRUCTION OF USE AND SHARED MEANING

Socio-technical design system

Institutionalised worlds of consumption Family consumption style Peers' consumption style SUBJECT intersubjective OTHERS space including focusing on the OBJECT cultural object or product -the institutionalised consumption arenas (i.e. distribution and retailing, regulatory requirements, socially admitted sales techniques, sales on Internet, cultural events like Christmas, Easter, Carnival festivities, etc.), and the ongoing innovativeness of the socio-technical system that bring new possibilities to design products for children. For example "smart toys" or interactive cuddly toys mix electronics and traditional techniques and reflect the development of the toy industry especially in relation to technological innovation. Within this system, some mediators developed by managers play a major role such as:

-Packaging: structures and channels children's learning experiences by constraining the categorisation process used to recognize, classify and compare products [START_REF] Cochoy | Une sociologie du packaging ou l'âne de buridan face au marché[END_REF]) -Brand characters: creates an affective relationship between the brand and the child and establishes brand recognition by young consumers [START_REF] Lawrence | The role of characters in kids marketing[END_REF]) -Advertising: the mass media (Kline, 1993 and2003 ;[START_REF] Tufte | Children, media and consumption[END_REF] contribute to the diffusion of values, social norms, symbolic languages and behaviours linked to consumption practices.

As a consequence, children's consumption practices cannot be reduced to a mere purchase decision: educational, social and institutional dimensions are at the core of the long process allowing children to take part in consumption realms. Moreover, when using an item, the child discovers and learns important information about social life and its normative requirements. Thus, child consumption is both a mediated and a mediating social activity through which the child learns many things far beyond consumption itself [START_REF] De La Ville | L'activité de consommation enfantine et ses médiateurs[END_REF]. When choosing a gift for a birthday party, the child learns that different social rules have to be called on if the beneficiary is a boy, a girl, or an adult, etc. (Cook, 2002).

Learning to consume: moving from peripheral to central participation

Considering this complex system, the central focus of inquiry becomes: how do children participate in socio-cultural consumption activities and how does their position change from being peripheral participants [START_REF] Lave | Situated Learning, Legitimate Peripheral Participation[END_REF] -as users of a product bought by their parents, as observers of buying situations…-, to central participants -as decision-makers on some categories of food products, as recognized experts of some specific products? Indeed, learning to consume is not simply a matter of socialisation: it supposes the child is able to adjust his level of participation in the activity depending on varied socio-cultural rules [START_REF] Rogoff | [END_REF] In this perspective, research should focus on deciphering the variety of consumption activities in which children are involved and on understanding the social and cultural conditions that allow them to adjust their participation to different degrees: 1°) a peripheral participation -where the young children are mainly in a position of observers -when discovering the items bought back home by their parents or older siblings or when accompanying their parents to shop in traditional markets, retail outlets or when shopping on-line at home ; 2°) a more focused and active participation of children through direct and indirect prescription concerning the items about which their caregivers, peers or siblings consider, value, and legitimate their desire and point of view; 3°) a central participation, where children act as active and competent consumers, able to legitimately decide how to spend their own pocket money in some consumption areas that are culturally defined as specific spaces of childhood within a cultural community.

This cultural perspective highlights that children are able to create their own social rules and to modify words and objects belonging to adulthood. In doing so, they generate their own socially shared meanings and cultures. The heroes and celebrities they admire, the new forms of narratives they explore, the games and playing activities they are involved in, the possibilities brought by the technologies they use, etc. are constitutive elements of children's culture that differs from their parents' one [START_REF] Buckingham | After the Death of Childhood[END_REF]. Birthday celebrations are a very emblematic form of children's socialisation where the convenient social rules to be followed are not defined only by parents but by children themselves [START_REF] Sirota | Les civilités de l'enfance contemporaine : l'anniversaire ou le déchiffrage d'une configuration[END_REF]Cook et al., 2002).

Future Research Directions

We have contrasted and clarified the potential contributions of three perspectives of developmental psychology to the understanding of how children gradually enter the realm of consumption practice and learn to take part in varied consumption activities. Considering socialisation itself as a fundamentally solitary cognitive construction by an individual developing through a succession of stages, researchers in management and marketing tend to perpetuate a quite restricted understanding of consumer socialisation of the child, privileging the logico-mathematical dimension of thought and overlooking the socio-cultural and historical dimensions constitutive of consumption. Thus, there still exist gaps in our understanding of the various factors that are inter-related in the process consumer socialisation: "Cultural changes, such as the growth of single-parent families, and technological change, such as the internet, suggest the need to revisit existing findings about socialization and address new concerns." (Roedder-John, 2002: 79).

In order to enhance a comprehensive perspective on how children elaborate their reasoning about consumption activities, marketing research should focus on the syntagmatic language children practise in their daily life. The consumption activities in which children are involved constitute a 'permanent recreation', an ongoing invention through which children renew cultural assets -such as products, toys, etc. for instance -by transforming them, adapting them to particular circumstances, recombining them according to the perceived stakes, and inventing meanings that remain temporary, depending on technological and media advances, in particular through websites dedicated to information and commerce. Thus, from a methodological standpoint, this inventive dimension of child consumption practices should lead to not considering the child as a mere respondent, but as a co-researcher who actively participates in the interpretation of his own experiences as an apprentice consumer [START_REF] Woodhead | Subjects, Objects or Participants? Dilemmas of psychological research with children[END_REF]Greene and Hogan, 2005).

A second line of investigation clearly includes the issue of risks linked to consumption, as well as the question of how the child's desires are considered in the daily activities in which he or she participates. As a consequence, another key research perspective could be aimed at understanding the modes of informal education -within retail stores, at home, among siblings or in peer groups -that contribute to shaping children's knowledge about consumption practices in relation to specific cultural contexts. It should lead to a reflection on the status of the child in the consumption society and the methods of setting up education for consumption aimed at increasing the degree of vigilance and reflexivity of the young consumer, and also contribute to specifying the consumption situations in which the child may be considered a vulnerable consumer in need of protection.

A third line of improvement includes the analysis of the historical, institutional and cultural aspects of joint-problem solving activities that are observed in consumption practices. In socio-cultural approaches, and pay special attention to how participants and institutions both determine the goals, means, situation, and definition of consumption activities. A reflection about theevolution of the institutional elements that enable and justify the child's position right at the heart of commercial situations is essential as it underlines that children's economic socialisation takes place within a broader historical trend in Western cultures appeared since the beginning of the XX th century in the USA -the 'empowerment' of the child as a consumer in his own right -that provides young consumers with institutional, cultural, and ideological resources to legitimate their active participation in the marketplace [START_REF] Cook | The commodification of childhood -The children's clothing industry and the rise of the child consumer[END_REF].

Discussion Questions

 To introduce three perspectives used in marketing research to explain how children develop as consumers :  dichotomy opposing the 'competent child' to the 'vulnerable child'

Figure 3 :

 3 Figure 3: Consumer socialisation as a mediating and mediated cultural experience


Ben Abdelkader

Sami Hassine

Othmen Mabrouk

Ncib

Abdelkader Ben Hassine
email: benhassine.abdelkader@yahoo.fr

Sami Mabrouk
email: mabrouksami00@yahoo.fr

Othmen Ncib
email: othmenncib@yahoo.fr

SOME CONSTRUCTIONS OF MULTIPLICATIVE n-ARY HOM-NAMBU ALGEBRAS

Keywords: 2000 Mathematics Subject Classification. 17A30, 17A36, 17A40, 17A42 n-ary Nambu algebra, Hom-Lie triple system, Hom-Lie n-uplet system, derivations, quasiderivations

We show that given a Hom-Lie algebra one can construct the n-ary Hom-Lie bracket by means of an (n-2)-cochain of given Hom-Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, there by inducing the structure of n-Hom-Lie algebra. We introduce the notion of Hom-Lie n-uplet system which is the generalization of Hom-Lie triple system. We construct Hom-Lie n-uplet system using a Hom-Lie algebra.

Introduction

The first instances of n-ary algebras in Physics appeared with a generalization of the Hamiltonian mechanics proposed in 1973 by Nambu [START_REF] Nambu | Generalized Hamiltonian mechanics[END_REF]. More recent motivation comes from string theory and Mbranes involving naturally an algebra with ternary operation called Bagger-Lambert algebra which gives impulse to a significant development. It was used in [START_REF] Bagger | Gauge symmetry and supersymmetry of multipleM2-branes[END_REF] as one of the main ingredients in the construction of a new type of supersymmetric gauge theory that is consistent with all the symmetries expected of a multiple M2-brane theory: 16 supersymmetries, conformal invariance, and an SO(8) R-symmetry that acts on the eight transverse scalars. On the other hand in the study of supergravity solutions describing M2-branes ending on M5-branes, the Lie algebra appearing in the original Nahm equations has to be replaced with a generalization involving ternary bracket in the lifted Nahm equations(see [START_REF] Basu | The M2-M5 brane system and a generalized Nahm's equation[END_REF]).

In [START_REF] Ataguema | Generalization of n-ary Nambu algebras and beyond[END_REF], generalizations of n-ary algebras of Lie type and associative type by twisting the identities using linear maps have been introduced. The notion of representations, derivations, cohomology and deformations are studies in [START_REF] Arnlind | Structure and cohomology of 3-Lie algebras induced by Lie algebras[END_REF][START_REF] Daletskii | Leibniz and Lie algebra structures for Nambu algebra[END_REF][START_REF] Ph | Some remarks concerning Nambu mechanics[END_REF][START_REF] Ma | On the cohomology and extensions of first-class n-Lie superalgebras[END_REF][START_REF] Takhtajan | On foundation of the generalized Nambu mechanics[END_REF]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-Lie algebra (called also n-ary Nambu-Lie algebras) and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. In [START_REF] Arnlind | Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras[END_REF], a method was demonstrated of how to construct ternary multiplications from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras. This construction was generalized to n-Lie algebras and n-Hom-Nambu-Lie algebras in [START_REF] Arnlind | Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras[END_REF].

It is well known that the algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. The notion of δ-derivations appeared in the paper of Filippov [START_REF] Filippov | δ-Derivations of Lie algebras[END_REF]. The results for δ-derivations and generalized derivations were studied by many authors. For example, R.Zhang and Y.Zhang [START_REF] Zhang | Generalized derivations of Lie superalgebras[END_REF] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, Hom-Lie superalgebras and Lie triple systems [START_REF] Chen | Generalized Derivations of Lie color algebras[END_REF][START_REF] Chen | Generalized Derivations of Lie triple systems[END_REF]. Derivations and generalized derivations of n-ary algebras were considered in [START_REF] Kaygorodov | δ-Derivations of n-ary algebras[END_REF][START_REF] Kaygorodov | n + 1)-ary derivations of semisimple Filippov algebras[END_REF] and other. In [START_REF] Beites | Generalized derivations of multiplicative n-ary Hom-Ω color algebras[END_REF], the authors generalize this results in Color n-ary Nambu Hom case. This paper is organized as follows. In Section I we review basic concepts of Hom-Lie, n-ary Hom-Nambu algebras and n-Hom-Lie algebras. We also recall the definitions of derivations, α k -derivations, α k -quasiderivations and α k -centroid. In Section II we provide a construction procedure of n-Hom-Lie algebras starting from a binary bracket of a Hom-Lie algebra and multilinear form satisfying certain conditions. To this end, we give the relation between α k -derivations, (resp. α k -quasiderivations and α kcentroid) of Hom-Lie algebra and α k -derivations, (resp. α k -quasiderivations and α k -centroid) of n-Hom-Lie algebras. In Section III we introduce the notion of Hom-Lie n-uplet system which is the generalisation of Lie n-uplet system which is introduced in [START_REF] De Azcárraga | k-Leibniz algebras from lower order ones: From Lie triple to Lie ℓ-ple systems[END_REF]. We construct Hom-Lie n-uplet system using Hom-Lie algebra. Finally, we give a relation between α k -quasiderivations of a Hom-Lie algebra and (n + 1)-ary α k -derivation of Hom-Lie n-uplet system associate.

Hom-Lie algebra and n-ary Hom-Nambu algebras

Throughout this paper, we will for simplicity of exposition assume that K is an algebraically closed field of characteristic zero, even though for most of the general definitions and results in the paper this assumption is not essential.

1.1. Definitions. The notion of a Hom-Lie algebra was initially motivated by examples of deformed Lie algebras coming from twisted discretizations of vector fields(see [START_REF] Hartwig | Deformations of Lie algebras using σ-derivations[END_REF][START_REF] Larsson | Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities[END_REF]). We will follow notation conventions in [START_REF] Makhlouf | Hom-algebra structure[END_REF].

Definition 1.1. A Hom-Lie algebra is a triple (g, [ , ], α), where [ , ] : g × g → g is a bilinear map and α : g → g a linear map satisfying

[x, y] = -[y, x] (skew-symmetry)
x,y,z [α(x), [y, z]] = 0 (Hom-Jacobi condition) for all x, y ,z from g, where x,y,z denotes summation over the cyclic permutations of x, y, z.

Definition 1.2. A Hom-Lie algebras (g, [ , ], α) is called multiplicative if α([x, y]) = [α(x), α(y)] for all x, y ∈ g.
We define a linear map ad : g → End(g) by ad x (y) = [x, y]. Thus the Hom-Jacobi identity is equivalent to (1.1) ad [x,y] (α(z)) = ad α(x) • ad y (z)ad α(y) • ad x (z), for all x, y, z ∈ g.

Remark 1.3. An ordinary Lie algebras is a Hom-Lie algebas when α = id.

Example 1.4. Let A be the complex algebra where A = C[t, t -1 ] is the Laurent polynomials in one variable. The generators of A are of the form t n for n ∈ Z. Let q ∈ C\{0, 1} and n ∈ N, we set {n} = 1-q n 1-q , a q-number. The q-numbers have the following properties {n + 1} = 1 + q{n} = {n} + q n and {n + m} = {n} + q n {m}. Let A q be a space with basis {L m , I m |m ∈ Z} where L m = -t m D, I m = -t m and D is a q-derivation on A such that

D(t m ) = {m}t m .
We define the bracket [ , ] q : A q × A q -→ A q , with respect the super-skew-symmetry for n, m ∈ Z by

[L m , L n ] q = ({m} -{n})L m+n , (1.2) [L m , I n ] q = -{n}I m+n , (1.3) [I m , I n ] q = 0. (1.4)
Let α be an even linear map on A q defined on the generators by

α q (L n ) = (1 + q n )L n , α q (I n ) = (1 + q n )I n ,
The triple (A q , [ , ] q , α q ) is a Hom-Lie algebra, called the q-deformed Heisenberg-Virasoro algebra of Hom-type.

Example 1.5. We consider the matrix construction of the algebra sl 2 (R) generated by the following three vectors

H = 1 0 0 -1 ; X = 0 1 0 0 ; Y = 0 0 1 0 The defining relations are [H, X] = 2X ; [H, Y ] = -2Y ; [X, Y ] = H Let λ ∈ R * , consider the linear maps α λ : sl 2 (R) → sl 2 (R) defined by α λ (H) = H ; α λ (X) = λ 2 X ; α λ (Y ) = 1 λ 2 Y Note that α λ is a Lie algebra automorphism.
In [START_REF] Ammar | Hom-Lie algebras and Hom-Lie admissible superalgebras[END_REF], the authors have shown that (sl 2 (R)) λ = (sl 2 (R), [ , ] α λ , α λ ) is a family of multiplicative Hom-Lie algebras where the Hom-Lie bracket [ , ] α λ on the basis elements is given, for λ = 0 by

[H, X] α λ = 2λ 2 X ; [H, Y ] α λ = - 2 λ 2 Y ; [X, Y ] α λ = H
Now, we recall the definitions of n-ary Hom-Nambu algebras and n-ary Hom-Nambu-Lie algebras, generalizing of n-ary Nambu algebras and n-ary Nambu-Lie algebras (called also Filippov algebras) respectively, which were introduced in [START_REF] Ataguema | Generalization of n-ary Nambu algebras and beyond[END_REF] by Ataguema, Makhlouf and Silvestrov. Definition 1.6. An n-ary Hom-Nambu algebra is a triple (N , [ , ..., ], α) consisting of a vector space N , an n-linear map [ , ..., ] : N n -→ N and a family α = (α i ) 1≤i≤n-1 of linear maps α i : N -→ N , satisfying

α 1 (x 1 ), ...., α n-1 (x n-1 ), [y 1 , ...., y n ] = (1.5) n i=1 α 1 (y 1 ), ...., α i-1 (y i-1 ), [x 1 , ...., x n-1 , y i ], α i (y i+1 ), ..., α n-1 (y n ) , for all (x 1 , ..., x n-1 ) ∈ N n-1 , (y 1 , ..., y n ) ∈ N n . The identity (1.5) is called Hom-Nambu identity. Let X = (x 1 , . . . , x n-1 ) ∈ N n-1 , α(X) = (α 1 (x 1 ), . . . , α n-1 (x n-1
)) ∈ N n-1 and y ∈ N . We define an adjoint map ad(X) as a linear map on N , such that

(1.6) ad X (y) = [x 1 , • • • , x n-1 , y].
Then the Hom-Nambu identity (1.5) may be written in terms of adjoint map as

ad α(X) ([x n , ..., x 2n-1 ]) = 2n-1 i=n [α 1 (x n ), ..., α i-n (x i-1 ), ad X (x i ), α i-n+1 (x i+1 )..., α n-1 (x 2n-1 )]. Definition 1.7. An n-ary Hom-Nambu algebra is a triple (N , [ , ..., ], α) is called n-Hom-Lie algebra if the bracket [ , ..., ] is skewsymmetric i.e [x σ(1) , • • • , x σ(n) ] = (-1) sign(σ) [x 1 , • • • , x n ] for σ ∈ S n .
Remark 1.8. When the maps (α i ) 1≤i≤n-1 are all identity maps, one recovers the classical n-ary Nambu algebras. The Hom-Nambu identity (1.5), for n = 2, corresponds to Hom-Jacobi identity (see [START_REF] Makhlouf | Hom-algebra structure[END_REF]), which reduces to Jacobi identity when α 1 = id.

Let (N , [ , . . . , ], α) and (N ′ , [•, . . . , •] ′ , α ′ ) be two n-ary Hom-Nambu algebras where α = (α i ) i=1,••• ,n-1 and α ′ = (α ′ i ) i=1,••• ,n-1 . A linear map f : N → N ′ is an n-ary Hom-Nambu algebras morphism if it satisfies f ([x 1 , • • • , x 2n-1 ]) = [f (x 1 ), • • • , f (x 2n-1 )] ′ f • α i = α ′ i • f ∀i = 1, n -1.
In the sequel we deal sometimes with a particular class of n-ary Hom-Nambu algebras which we call n-ary multiplicative Hom-Nambu algebras. Definition 1.9. A multiplicative n-ary Hom-Nambu algebra (resp. multiplicative n-Hom-Lie algebra) is an n-ary Hom-Nambu algebra (resp. n-Hom-Lie algebra) (N , [ , ..., ], α) with α = (α i ) 1≤i≤n-1 where α 1 = ... = α n-1 = α and satisfying

(1.7) α([x 1 , .., x n ]) = [α(x 1 ), .., α(x n )], ∀x 1 , ..., x n ∈ N .
For simplicity, we will denote the n-ary multiplicative Hom-Nambu algebra as (N , [ , ..., ], α) where α : N → N is a linear map. Also by misuse of language an element x ∈ N n refers to x = (x 1 , .., x n ) and α(x) denotes (α(x 1 ), . . . , α(x n )).

1.2. Derivations, Quasiderivations and centroids of multiplicative n-Hom-Lie algebras.

In this section we recall the definition of derivation, generalized derivation, quasiderivation and centroids of multiplicative n-Hom-Lie algebras. Let (N , [ , • • • , ], α) be an multiplicative n-Hom-Lie algebra. We denote by α k the k-times composition of α (i.e.

α k = α • ... • α k-times). In particular α -1 = 0, α 0 = id. Definition 1.10. For any k ≥ 1, we call D ∈ End(N ) an α k -derivation of the multiplicative n-Hom-Lie algebra (N , [ , ..., ], α) if (1.8) [D, α] = 0 i.e. D • α = α • D,
and

(1.9) D[x 1 , ..., x n ] = n i=1 [α k (x 1 ), ..., α k (x i-1 ), D(x i ), α k (x i+1 ), ..., α k (x n )],
We denote by Der α k (N ) the set of α k -derivations of the multiplicative n-Hom-Lie algebra N .

For X = (x 1 , ..., x n-1 ) ∈ N n-1 satisfying α(X) = X and k ≥ 1, we define the map ad k X ∈ End(N ) by (1.10) ad k X (y) = [x 1 , ..., x n-1 , α k (y)] ∀y ∈ N . Then Lemma 1.11. The map ad k
X is an α k+1 -derivation, that we call inner α k+1 -derivation. We denote by Inn α k (N ) the space generate by all the inner α k+1 -derivations. For any D ∈ Der α k (N ) and D ′ ∈ Der α k (N ) we define their commutator [D, D ′ ] as usual:

(1.11) [D, D ′ ] = D • D ′ -D ′ • D. Set Der(N ) = k≥-1 Der α k (N ) and Inn(N ) = k≥-1 Inn α k (N ).
Definition 1.12. An endomorphism D of a multiplicative n-ary Hom-Nambu algebra (N , [ ,

• • • , ], α) is called a generalized α k -derivation if there exist linear mappings D ′ , D ′′ , • • • , D (n-1) , D (n) ∈ End(N ) such that (1.12) D (n) ([x 1 , • • • , x n ]) = n i=1 [α k (x 1 ), • • • , D (i-1) (x i ), • • • , α k (x n )], for all x 1 , • • • , x n ∈ N . An (n + 1)-tuple (D, D ′ , D ′′ , • • • , D (n-1) , D (n) ) is called an (n + 1)-ary α k - derivation.
The set of generalized α k -derivation is denoted by

GDer α k (N ). Set GDer(N ) = k≥-1 GDer α k (N ). Definition 1.13. Let (N , [ , • • • , ],
α) be a multiplicative n-ary Hom-Nambu algebra and End(N ) be the endomorphism algebra of N . An endomorphism D ∈ End(N ) is said to be an α k -quasiderivation, if there exists an endomorphism

D ′ ∈ End(N ) such that n i=1 [α k (x 1 ), . . . , D(x i ), . . . , α k (x n )] = D ′ ([x 1 , . . . , x n ]),
for all x 1 , . . . , x n ∈ N . We call D ′ the endomorphism associate to the α k -quasiderivation D.

The set of α k -quasiderivations will be denoted by QDer α k (N ). Set QDer(N ) = k≥-1

QDer α k (N ).

Cent α k (N ) = {θ ∈ End(N ) : θ[x 1 , • • • , x n ] = [θ(x 1 ), α k (x 2 ), . . . , α k (x n )], ∀x i ∈ N }.
We recover the definition of the centroid when k = 0.

If N is a multiplicative n-Hom-Lie algebra, then it is a simple fact that θ[x 1 , . . . , x n ] = [α k (x 1 ), . . . , θ(x p ), . . . , α k (x n )], ∀p ∈ {1, . . . , n}.

n-Hom-Lie algebras induced by Hom-Lie algebras

In [START_REF] Arnlind | Structure and cohomology of 3-Lie algebras induced by Lie algebras[END_REF] and [START_REF] Arnlind | Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras[END_REF], the authors introduced a construction of a 3-Hom-Lie algebra from a Hom-Lie algebra, and more generally of an (n+1)-Hom-Lie algebra from an n-Hom-Lie algebra. It is called (n+1)-Hom-Lie algebra induced by n-Hom-Lie algebra. In this context, Abramov give a new approach of this construction (see [START_REF] Abramov | Weil Algebra, 3-Lie Algebra and BRS Algebra[END_REF]). Now we generalise this approach in the Hom case.

Let (g, [ , ], α) a multiplicative Hom-Lie algebra and g * be its dual space. Fix an element of the dual space ϕ ∈ g * . Define the triple product as follows Now we consider ϕ as a K-valued cochain of degree one of the Chevalley-Eilenberg complex of a Lie algebra g. Making use of the coboundary operator δ :

∧ k g * → ∧ k g * defined by (2.3) δf (u 1 , • • • , u k+1 ) = i<j (-1) i+j+1 f ([u i , u j ] g , α(u 1 ) • • • , u i , • • • , u j , • • • , α(u k+1 )),
for f ∈ ∧ k g * and for all u 1 , • • • , u k+1 ∈ g. We obtain that δϕ(x, y) = ϕ([x, y]).

Finally we can define the wedge product of two cochains ϕ and δϕ, which is the cochain of degree three by ϕ ∧ δϕ(x, y, z) = ϕ(x)ϕ([y, z]) + ϕ(y)ϕ([z, x]) + ϕ(z)ϕ([x, y]). Hence (2.2) is equivalent to ϕ ∧ δϕ = 0. Thus if an 1-cochain ϕ satisfies the equation (2.2) then the triple product (2.1) is the ternary Lie bracket and we will call this multiplicative 3-Hom-Lie bracket the quantum Nambu bracket induced by a 1-cochain. Definition 2.1. Let φ ∈ ∧ n-2 g * , we define the n-ary product as follows

(2.4) [x 1 , • • • , x n ] φ = n i<j (-1) i+j+1 φ(x 1 , • • • , xi , • • • , xj , • • • , x n )[x i , x j ], for all x 1 , • • • , x n ∈ g.
Then, we obtain.

Proposition 2.2. The n-ary product [ , • • • , ] φ is skew-symetric.

Proof. Let x 1 , • • • , x n ∈ g and fixed two integer i < j, we have:

[x 1 , • • • , x i , • • • , x j , • • • , x n ] φ = k<l:k,l =i,j (-1) k+l+1 φ(x 1 , • • • , x i • • • , x k , • • • , x j , • • • , x l , • • • , x n )[x l , x k ] + i<l =j (-1) i+l+1 φ(x 1 , • • • , x i • • • , x j , • • • , x l , • • • , x n )[x i , x l ] + l<i (-1) i+l+1 φ(x 1 , • • • , x l , • • • , x i , • • • , x j , • • • , x n )[x l , x i ] + j<l (-1) j+l+1 φ(x 1 , • • • , x i • • • , x j , • • • , x l , • • • , x n )[x j , x l ] + l<j,i =l (-1) j+l+1 φ(x 1 , • • • , x i , • • • , x l , • • • , x j , • • • , x n )[x l , x j ] + (-1) i+j+1 φ(x 1 , • • • , x i , • • • , x j , • • • , x n )[x i , x j ] = -[x 1 , • • • , x j , • • • , x i , • • • , x n ] φ . Given X = (x 1 , • • • , x n-3 ) ∈ ∧ n-3 g, Y = (y 1 , • • • , y n ) ∈ ∧ n g
and z ∈ g. We define the linear map φ X by: φ X (z) = φ(X, z), and

φ ∧ δφ X (Y ) = n i<j (-1) i+j φ(y 1 , • • • ŷi • • • ŷj • • • , y n-1 )δφ X (y i , y j ) = n i<j (-1) i+j φ(y 1 , • • • ŷi • • • ŷj • • • , y n )φ X ([y i , y j ]).
Theorem 2.3. Let (g, [ , ], α) be a multiplicative Hom-Lie algebra, g * be its dual and φ be a cochain of degree n -2, i.e. φ ∈ ∧ n-2 g * . The vector space g equipped with the n-ary product (2.4) and the linear map α is a multiplicative n-Hom-Lie algebra if and only if

φ ∧ δφ X = 0, ∀X ∈ ∧ n-3 g, (2.5) φ • (α ⊗ Id ⊗ • • • ⊗ Id) = φ. (2.6) Proof. Firstly, let (x 1 , • • • , x n ) ∈ ∧ n g. We have: [α(x 1 ), • • • , α(x n )] φ = n i<j (-1) i+j+1 φ(α(x 1 ), • • • , α(x i ), • • • , α(x j ), • • • , α(x n ))[α(x i ), α(x j )] = n i<j (-1) i+j+1 φ(x 1 , • • • , xi , • • • , xj , • • • , x n )α([x i , x j ]) = α([x 1 , • • • , x n ] φ ). Secondly, for (x 1 , • • • , x n-1 ) ∈ ∧ n-1 g and (y 1 , • • • , y n ) ∈ ∧ n g, we have [α(x 1 ), • • • , α(x n-1 ), [y 1 , • • • , y n ] φ ] φ = i<j (-1) i+j+1 φ(y 1 , • • • , y i , • • • , y j , • • • , y n )[α(x 1 ), • • • , α(x n-1 ), [y i , y j ]] φ = i<j k<l≤n-1 (-1) i+j+k+l φ(α(x 1 ), • • • , α(x k ), • • • , α(x l ), • • • , [y i , y j ])φ(y 1 , • • • , y i , • • • , y j , • • • , y n )[α(x k ), α(x l )] + i<j k<n (-1) i+j+k φ(α(x 1 ), • • • , α(x k ), • • • , α(x (n-1) ), • • • , [y i , y j ])φ(y 1 , • • • , y i , • • • , y j , • • • , y n )[α(x k ), [y i , y j ]].
The terms [α(x k ), [y i , y j ]] are simplified by identity of Jacobi in the second half of the Filippov identity. Now, we group together the other terms according to their coefficient [α(x i ), α(x j )]. For example, if we fixed (k, l) and, if we collect all the terms containing the commutator [α(x k ), α(x l )], then we get the expression

i<j (-1) i+j+k+l φ(α(x 1 ), • • • , α(x k ), • • • , α(x l ), • • • , [y i , y j ])φ(y 1 , • • • , y i , • • • , y j , • • • , y n ) [α(x k ), α(x l )].
Hence the n-ary product (2.4) will satisfy the n-ary Filippov-Jacobi identity if for any elements

X = (x 1 , • • • , x n-3 ) ∈ ∧ n-3 g and Y = (y 1 , • • • , y n ) ∈ ∧ n g we require n i<j (-1) i+j φ(α(x 1 ), • • • , α(x n-3 ), [y i , y j ])φ(y 1 , • • • , y i , • • • , y j , • • • , y n ) = 0.
φ • (Id ⊗ • • • ⊗ Id ⊗ [ , ]) = 0 and φ • (α ⊗ Id ⊗ • • • ⊗ Id) = φ.
Corollary 2.5. Let φ : g ⊗n-2 → K be a trace of Hom-Lie algebra (g, [ , ], α), then g φ = (g, [., • • • , .] φ , α) is a n-Hom-Lie algebra.

Proposition 2.6. Let (g, [ , ], α) be a Hom-Lie algebra and D ∈ Der(g) be an α k -derivation such that

n-2 i=1 φ(x 1 , • • • D(x i ), • • • , x n-2 ) = 0. Then D is an α k -derivation of the n-Hom-Lie algebra (g, [ , • • • , ] φ , α). Proof. Let X = (x 1 , • • • , x n ) ∈ ∧ n g, on the one hand we get D([x 1 , • • • , x n ] φ ) = D i<j (-1) i+j+1 φ(α(x 1 ), • • • , α( x i ), • • • , α( x j ), • • • , α(x n ))[α(x i ), α(x j )] = i<j (-1) i+j+1 φ(α(x 1 ), • • • , α( x i ), • • • , α( x j ), • • • , α(x n ))D([α(x i ), α(x j )]) = i<j (-1) i+j+1 φ(x 1 , • • • , x i , • • • , x j , • • • , x n )[α(D(x i )), α k+1 (x j )] + i<j (-1) i+j+1 φ(x 1 , • • • , x i , • • • , x j , • • • , x n )[α k+1 (x i ), α(D(x j ))],
and on the other hand, we have

n l=1 [α k (x 1 ), • • • , α k (x l-1 ), D(x l ), • • • , α k (x l+1 ), • • • , α k (x n )] φ = n l=1 i<j ; i,j =l (-1) i+j+1 φ(α k (x 1 ), • • • , α k (x i ), • • • , D(x l ), • • • , α k (x j ), • • • , α k (x n ))[α k (x i ), α k (x j )] + n l=1 i<l (-1) i+l+1 φ(α k (x 1 ), • • • , α k (x i ), • • • , D(x l ), • • • , α k (x n ))[α k (x i ), D(x l )] + n l=1 l=i<j (-1) j+l+1 φ(α k (x 1 ), • • • , D(x l ), • • • , α k (x j ), • • • , α k (x n ))[D(x l ), α k (x j )]. If D is an α k -derivation then D([x 1 , • • • , x n ] φ ) = n l=1 [α k (x 1 ), • • • , α k (x l-1 ), D(x l ), • • • , α k (x l+1 ), • • • , α k (x n )] φ , which gives i<j ; i,j =l (-1) i+j+1 n l=1 φ(α k (x 1 ), • • • , α k (x i ), • • • , D(x l ), • • • , α k (x j ), • • • , α k (x n )) [α k (x i ), α k (x j )] = 0. Finally if we fixed (i, j) we have n-2 l=1 φ(α k (x 1 ), • • • , D(x l ), • • • , α k (x n-2 )) = 0.
Proposition 2.7. Let (g, [ , ], α) be a Hom-Lie algebra and D ∈ QDer(g) be an α k -quasiderivation and D ′ : g → g the endomorphism associate to D such that

n-2 i=1 φ(x 1 , • • • D(x i ), • • • , x n-2 ) = 0.
Then D is an α k -quasiderivation of the n-Hom-Lie algebra (g, [ , • • • , ] φ , α) with the same endomorphism associate D ′ . Proposition 2.8. Let (g, [ , ], α) be a Hom-Lie algebra and θ : g → g be an α k -centroid such that

φ(θ(x 1 ), • • • x i , • • • , x n-2 )[α k (x), y] = φ(x 1 , • • • x i , • • • , x n-2 )[θ(x), y]. Then D is an α k -centroid on the n-Hom-Lie algebra (g, [ , • • • , ] φ , α). Proof. Let x 1 , • • • , x n ∈ g, we have θ([x 1 , • • • , x n ] φ ) = n i<j (-1) i+j+1 φ(x 1 , • • • , xi , • • • , xj , • • • , x n )θ([x i , x j ]) = n i<j (-1) i+j+1 φ(x 1 , • • • , xi , • • • , xj , • • • , x n )[θ(x i ), α k (x j )].
On the other hand, we have

[θ(x 1 ), α k (x 2 ), • • • , α k (x n )] φ = n i<j (-1) i+j+1 φ(θ(x 1 ), α k (x 2 ), • • • , xi , • • • , xj , • • • , α k (x n ))[α k (x i ), α k (x j )] = n i<j (-1) i+j+1 φ(x 1 , • • • , xi , • • • , xj , • • • , x n )[θ(x i ), α k (x j )] = θ([x 1 , • • • , x n ] φ ).
Let (g, [ , ], α) be a Hom-Lie algebra. We define the following n linear map:

[ , • • • , ] n : g ⊗n -→ g (x 1 , • • • , x n ) -→ x 1 , • • • , x n n = [. . . [x 1 , x 2 ], α(x 3 )], α 2 (x 4 ) • • • α n-3 (x n-1 )], α n-2 (x n ) (3.1) For n=2, [x 1 , x 2 ] 2 = [x 1 , x 2 ] and for n ≥ 3 we have [x 1 , • • • , x n ] n = [[x 1 , • • • , x n-1 ] n-1 , x n ].
The following result gives a way to construct Hom-Lie n-uplet system starting from a classical Lie n-uplet system and algebra endomorphisms.

Proposition 3.10. Let (G, [ , • • • , ]) a Lie n-uplet system and α : G → G a linear map such that α([x 1 , • • • , x n ]) = [α(x 1 ), • • • , α(x n )]. Then (G, [ , • • • , ] α , α) be a Hom-Lie n-uplet system, where [x 1 , • • • , x n ] α = [α(x 1 ), • • • , α(x n )], for all x 1 , • • • , x n ∈ G.
Theorem 3.11. Let (g, [ , ], α) be a multiplicative Hom-Lie algebra. Then

g n = (g, [ , • • • , ] n , α n-1 )
is a multiplicative Hom-Lie n-uplet system.

When n = 3 we obtain the multiplicative Hom-Lie triple system constructed in Theorem 3.5. To prove this theorem we need to the following lemma. Lemma 3.12. Let (g, [ , ], α) be a multiplicative Hom-Lie algebra, and ad 2 the adjoint map defined by ad 2

x (y) = ad x (y) = [x, y] Then, we have

ad 2 α n-1 (x) [y 1 , • • • , y n ] n = n k=1 [α(y 1 ), • • • , α(y k-1 ), ad 2 x (y k ), α(y k+1 ), • • • , α(y n )] n ,
where x ∈ g, y ∈ g and (y

1 , • • • , y n ) ∈ g n .
Proof. For n = 2, using the Hom-Jacobi identity we have x (y), α(z)] + [α(y), ad 2 x (z)]. Assume that the property is true up to order n, that is

ad 2 α n-1 (X) [y 1 , • • • , y n ] n = n k=1 [α(y 1 ), • • • , α(y k-1 ), ad 2 X (y k ), α(y k+1 ), • • • , α(y n )] n . Let x ∈ g and (y 1 , • • • , y n+1 ) ∈ g n+1 , we have ad 2 α n (x) [y 1 , • • • , y n+1 ] = ad 2 α n (x) [[y 1 , • • • , y n ] n , α n-1 (y n+1 )] 2 = ad 2 α n-1 (x) [y 1 , • • • , y n ] n , α n (y n+1 ) 2 + [α(y 1 ), • • • , α(y n )] n , ad 2 α n-1 (x) (α n-1 (y n+1 )) 2 = n k=1 [α(y 1 ), • • • , α(y k-1 ), ad 2 x (y k ), α(y k+1 ), • • • , α(y n )] n , α n (y n+1 ) + [α(y 1 ), • • • , α(y n )] n , α n-1 (ad 2 x (y n+1 )) 2 = n k=1 α(y 1 ), • • • , α(y k-1 ), ad 2 x (y k ), α(y k+1 ), • • • , α(y n ), α(y n+1 ) n+1 + α(y 1 ), • • • , α(y n ), ad 2 x (y n+1 ) n+1 = n+1 k=1 α(y 1 ), • • • , α(y k-1 ), ad 2 x (y k ), α(y k+1 ), • • • , α(y n+1 ) n+1 .
The lemma is proved.

Proof. (Proof of Theorem 3.11)

Let X = (x 1 , • • • , x n-1 ) ∈ g n-1 and Y = (y 1 , • • • , y n ) ∈ g n . (i) Its easy to see that [x 1 , x 1 , x 2 , • • • , x n-1 ] n = [[• • • [[x 1 , x 1 ] 2 , α(x 2 )] 2 , α 2 (x 3 )] 2 , • • • ] 2 , α n-2 (x n-1 )] 2 = 0
(ii) Using lemma (3.12), we have

α n-1 (x 1 ), • • • , α n-1 (x n-1 ), [y 1 , • • • , y n ] n n = [α n-1 (x 1 ), • • • , α n-1 (x n-1 )] n-1 , [α n-2 (y 1 ), • • • , α n-2 (y n )] n 2 = ad 2 α n-1 [x1,••• ,xn-1] ([α n-2 (y 1 ), • • • , α n-2 (y n )] n ) = n k=1 α n-1 (y 1 ), • • • , ad 2 [x1,••• ,xn-1] (α n-2 (y k )), • • • , α n-1 (y n ) n = n k=1 α n-1 (y 1 ), • • • , [[x 1 , • • • , x n-1 ], α n-2 (y k )] 2 , • • • , α n-1 (y n ) n = n k=1 α n-1 (y 1 ), • • • , [x 1 , • • • , x n-1 , y k ] n , • • • , α n-1 (y n ) n .
Example 3.13. Using Exemple 1.5 and by the theorem (3.11), for λ ∈ R * we have:

For n = 3, (sl 2 (R), [ , , ] 3 , α 2 λ ) is a Hom-Lie triple system. The different brackets are as follows:

[H, X, Y ] 3 = [[H, X] α λ , α λ (Y )] α λ = 2H ; [H, X, H] 3 = -4λ 4 X ; [H, Y, X] 3 = 4H. [H, Y, H] 3 = -4 λ 4 Y ; [X, Y, Y ] 3 = -2 λ 4 Y ; [X, Y, X] 3 = 2λ 4 X.
The other brackets are zeros.

For n = 4, (sl 2 (R), [ , , , ] 4 , α 3 λ ) is a 4-uplet Hom-Lie triple system. The different brackets are defined as follows: Now, suppose that the property is true to order n -1, i.e: ) is an (n + 1)-ary α k -derivation of g n .

[H, X, H, H] 4 = [[H, X, H] 3 , α 2 (H)] α λ = -4λ 4 [X, H] α λ = 8λ 6 X ; [H, X, H, Y ] 4 = -4H [H, Y, H, H] 4 = -8 λ 6 Y ; [H, Y, H, X] 4 = 4H ; [H, X, Y, X] 4 = 4λ 6 X ; [H, X, Y, Y ] 4 = -2 λ 6 Y [H, Y, X, X] 4 = 8λ 6 X ; [H, Y, X, Y ] 4 = -8 λ 6 Y ; [X, Y, X, Y ] 4 = 2H [X, Y, X, H] 4 = -4λ 6 X ; [X, Y, Y, X]
D([x 1 , • • • , x n-1 ] n-1 ) = n i=1 [α k (x 1 ), • • • , D(x k ), • • • , α k (x n-1 )] n-1 . Let (x 1 , • • • , x n ) ∈ g n , then D([x 1 , • • • , x n ] n ) = D( [x 1 , • • • , x n-1 ] n-1 , α n-2 (x n ) ) = D([x 1 , • • • , x n-1 ] n-1 ), α n+k-2 (x n ) + [α k (x 1 ), • • • , α k (x n-1 )] n-1 , D(α n-2 (x n )) = D([x 1 , • • • , x n-1 ] n-1 ), α n-2 (α k (x n )) + [α k (x 1 ), • • • , α k (x n-1 )] n-1 , α n-2 (D(x n )) = n-1 i=1 [α k (x 1 ), • • • , D(x i ), • • • , α k (x n-1 )] n-1 , α n-2 (α k (x n )) + [α k (x 1 ), • • • , α k (x n-1 ), D(x n )] n = n-1 i=1 [α k (x 1 ), • • • , D(x i ), • • • , α k (x n-1 ), α k (x n )] n + [α k (x 1 ), • • • , α k (x n-
Proof. Let x 1 , • • • , x n ∈ g, then

D (n-1) ([x 1 , • • • , x n ] n ) = D (n-1) ([[x 1 , • • • , x n-1 ] n-1 , x n ]) = [D (n-2) ([x 1 , • • • , x n-1 ] n-1 ), α k (x n )] + [[α k (x 1 ), • • • , α k (x n-1 )] n-1 , D (n-2) (x n )] . . . = [D(x 1 ), α k (x 2 ), • • • , α k (x n )] n + [α k (x 1 ), D(x 2 ), • • • , α k (x n )] n + [α k (x 1 ), α k (x 2 ), D ′ (x 3 ), • • • , α k (x n )] n + • • • + [α k (x 1 ), • • • , α k (x n-1 ), D (n-2) (x n )] n )
Therefore (n + 1)-tuple (D, D, D ′ , D ′′ , • • • , D (n-1) ) is an (n + 1)-ary α k -derivation of g n .

Definition 1 . 14 .Definition 1 . 15 .

 114115 Let (N , [ , • • • , ], α) be a multiplicative n-ary Hom-Nambu algebra and End(N ) be the endomorphism algebra of N . Then the following subalgebra of End(N )Cent(N ) = {θ ∈ End(N ) : θ([x 1 , . . . , x n ]) = [θ(x 1 ), . . . , x n ], ∀x i ∈ N }is said to be the centroid of the n-ary Hom-Nambu algebra. The definition is the same for classical case of n-ary Nambu algebra. We may also consider the same definition for any n-ary Hom-Nambu algebra. Now, let (N , [ , • • • , ], α) be a multiplicative n-ary Hom-Nambu algebra. An α k -centroid of a multiplicative n-ary Hom-Nambu algebra (N , [ , • • • , ], α) is a subalgebra of End(N ) denoted Cent α k (N ), given by

(2. 1 )

 1 [x, y, z] = ϕ(x)[y, z] + ϕ(y)[z, x] + ϕ(z)[x, y], ∀ x, y, z ∈ g. Obviously this triple product is skew-symmetric. Straightforward computation of the left hand side and the right hand side of the Filippov-Jacobi identity (1.5) if ϕ • α = ϕ and (2.2) ϕ(x)ϕ([y, z]) + ϕ(y)ϕ([z, x]) + ϕ(z)ϕ([x, y]) = 0.

Definition 2 . 4 .

 24 Let φ : g ⊗ • • •⊗ g → K a skew-symmetric multilinear form of the multiplicative Hom-Lie algebras (g, [ , ], α), then φ is called trace if:

ad 2 α

 2 (x) [y, z] = [α(x), [y, z]] = [[x, y], α(z)] + [α(y), [x, z]] = [ad 2

4 =

 4 2H ; [X, Y, Y, H] 4 = -4 λ 6 Y .The other brackets are zeros. Proposition 3.14. Let (g, [ , ], α) be a multiplicative Hom-Lie algebra and D : g → g an α k -derivation of g for an integer k. Then D is an α k -derivation of g n .Proof. By recurrenceFor n = 3. Let x, y, z ∈ g, we haveD([x, y, z]) = D([[x, y], α(z)]) = [D([x, y]), α k+1 (z)] + [[α k (x), α k (y)], D(α(z))] = [[D(x), α k (y)], α k+1 (z)] + [[α k (x), D(y)], α k+1 (z)] + [[α k (x), α k (y)], α(D(z))] = [D(x), α k (y), α k (z)] + [α k (x), D(y), α k (z)] + [α k (x), α k (y), D(z)]

  1 ), D(x n )] n = n i=1 [α k (x 1 ), • • • , D(x i ), • • • , α k (x n-1 ), α k (x n )] nProposition 3.15. Let (g, [ , ], α) be a multiplicative Hom-Lie algebra and D, D ′ , • • • , D (n-1) an endomorphisms of g such that D (i) is an α k -quasiderivation with associated endomorphism D (i+1) for 0 ≤ i ≤ n -2. Then the (n + 1)-tuple (D, D, D ′ , D ′′ , • • • , D (n-1)

Acknowledgments

We would like to thank Abdenacer Makhlouf and Viktor Abramov for helpful discussions and for their interest in this work.

3. Hom-Lie n-uplet system 3.1. Hom-Lie triple system. In this section, we start by recalling the definitions of Lie triple systems and Hom-Lie triple systems. Definition 3.1. ( [START_REF] Lister | A structure theory of Lie triple systems[END_REF]) A vector space T together with a trilinear map (x, y, z)

]) A Hom-Lie triple system (Hom-LTS for short) is denoted by (T, [•, •, •], α), which consists of an K-vector space T , a trilinear product [•, •, •] : T × T × T → T , and a linear map α : T → T , called the twisted map, such that α preserves the product and for all x, y, z, u, v ∈ T , (

Remark 3.3. When the twisted map α is equal to the identity map, a Hom-LTS is a LTS. So LTS are special examples of Hom-LTS.

for all x, y, z ∈ T .

is a multiplicative Hom-Lie triple system.

3.2.

Hom-Lie n-uplet system.

In this section we introduce the definitions of Lie n-uplet systems and multiplicative Hom-Lie n-uplet systems. We give the analogue of Theorem 3.5 in the Hom-Lie n-uplet systems case.

Definition 3.6. A vector space G together with a n-linear map (x

Remark 3.9. When the twisted maps α i is equal to the identity map, a Hom-Lie n-uplet systems is a Lie n-uplet systems. So Lie n-uplet systems are special examples of Hom-Lie n-uplet systems.


domain: math shs

text:
Carlotta Soldano
email: carlotta.soldano@gmail.com

Ferdinando Arzarello
email: ferdinando.arzarello@unito.it

Learning with the logic of inquiry: game-activities inside Dynamic Geometry Environments

Keywords: DGE game-activity, played-game, reflected-game, inquiring and justifying processes

The paper describes a game-activity proposed to 7 th grade students with the goal to make them discover the geometric property concerning the mutual relationship between two circles. The activity, called "the game of the two circles", is composed of a strategic game that students play in GeoGebra and an investigative task which requires conjecturing and generalization. The aim of the activity is to trigger an approach to mathematics based on the logic of inquiry. We analyse students' dialogues and actions paying particular attention to the additional values the game confers to the more traditional exploratory activities with dynamic geometry software.

Introduction

Many studies in mathematics education have documented the importance of making students explore mathematical situations before asking them to construct proofs [START_REF] Boero | Some dynamic mental processes underlying producing and proving conjectures[END_REF][START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed?[END_REF]. The exploration triggers the formulation and the checking of conjectures, introducing students into logical ways of reasoning. As pointed out by Dewey, all forms of logics, included the deductive logic, are consequence of inquiry processes: all logical forms (with their characteristic properties) arise within the operation of inquiry and are concerned with control of inquiry so that it may yield warranted assertions. This conception implies much more than that logical forms are disclosed or come to light when we reflect upon processes of inquiry that are in use. Of course it means that; but it also means that the forms originate in operations of inquiry. (Dewey, 1938, p.3,4) [START_REF] Boero | Some dynamic mental processes underlying producing and proving conjectures[END_REF] have observed the possibility of a cognitive continuity between the processes of discovering and justifying. It occurs when the students, in the construction of the proof, exploit the argumentations employed for producing the conjecture. [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed?[END_REF] has distinguished between a cognitive unity in the referencial system and in its structure. The first occurs if some expressions, drawings, or theorems used in the proof have already been used in the argumentation for supporting the conjecture. The second occurs if the inferences produced in the argumentation and in the proof are connected through the same structure (abduction, induction, deduction). [START_REF] Hintikka | Inquiry as inquiry: a logic of scientific discovery[END_REF], an eminent Finnish logician, analyzing Sherlock Holmes way of reasoning in inquiry processes, showed that the clever deductions he made are obtained by reversing the abductions (Peirce, 1960) produced while investigating. His works demonstrates the existence of an epistemic unity between inquiring and justifying processes.

In this study, we favor and emphasize the possibility of connections between the discovery and justifying processes by introducing strategic games within Dynamic Geometry Environments (DGEs): these are games in which players have to make strategic choices meant for setting up and coordinating actions aimed at the achievement of a goal. As it is known from the literature [START_REF] Arzarello | A cognitive analysis of dragging practices in Cabri environments[END_REF][START_REF] Baccaglini-Frank | Generating conjectures in dynamic geometry: The maintaining dragging model[END_REF], DGEs are particularly apt for triggering inquiring processes. Our conjecture is that in virtue of the game, the conjectures and abductions produced inside the DGE are not left isolated but can be connected together and reorganized in logical chains. In fact, for making a strategic choice within a game situation, players reflect backward on the moves made and forward on the possible moves to make. These reflections can support the construction of logical links through which reorganize the geometrical invariants observed during the players' moves. In fact, since the moves are made on dynamic figures and involve geometric elements, we wish that the strategic reflections made on the moves could affect also the geometric elements involved in the moves. For this reason, DGE game-activities can promote a kind of thinking which is different from the one triggered by more traditional explorative activities with DGS.

Theoretical framework

The interrogative logic or logic of inquiry, introduced by Jaako Hintikka (1999), proposes a back to the origin consideration of the discipline. According to Hintikka, the modern logic switched from the study of excellence in reasoning to the study of infallibility in reasoning: "preserving one's logical virtue becomes a more important concern than developing virtuosity in drawing logical inferences" (Hintikka, 1999, p.28). The rules of inference are definitory rules, which inform us about the possible inferences, but do not say anything about which inferences are appropriate in the current moment, which are not so and which ones are better than others. These types of considerations are the concern of strategic principles.

Hintikka conceives the process of seeking new knowledge as an interrogative game, which is a twoplayer game between an inquirer, who asks the questions and an oracle, who answers him. Observations can be thought of as answers put to an environment, a controlled experiments, a database stored in the memory of a computer, a diagnostic handbook, a witness in a court of law, or one's own tacit knowledge partly based on one's memory can be considered as questions asked to nature. "Strategies of questioning play a central role in interrogative games, these include strategies of information seeking by means of different choices of questions to be asked and of the order in which they are asked." (Hintikka, 1999, p.34).

Hintikka models the processes of verification and falsifications through a semantical game [START_REF] Hintikka | The principles of mathematics revisited[END_REF], which is a two-player game between a verifier, whose goal is to show the truth of a mathematical formula or statement and a falsifier, whose goal is to confute it. In order to establish the truth of the mathematical formula ∀𝑥 ∃𝑦 | 𝑆[𝑥, 𝑦] it is possible to imagine a game in which the falsifier choses a value x0 "in the most unfavorable way as far as the interests of the verifier are concerned" and the verifier should find a value y0 for y such that 𝑆[𝑥 0 , 𝑦 0 ] is true. The formula is true if there exists a winning strategy for the verifier of the game, while it is false if there exists a winning strategy for the falsifier of the game.

In our study, taking inspiration from Hintikka's semantical game, we designed DGE game-activities in order to aid students in their discovery of geometric properties, through the game-play and the guiding questions. Analysing students' actions, we distinguish between two ways of using the game: the played-game and the reflected-game [START_REF] Soldano | Learning with touchscreen devices : A game approach as strategies to improve geometric thinking[END_REF]. In the played-game, the students' aim is to win against their opponent. To reach this goal they activate strategic principles which help them to select the best move to make in a given situation. In the reflected-game students play the game in order to answer the questionnaire and to communicate with each other. They play the game in a fictitious way: the game helps students to formulate the correct answer. In the reflected game we distinguish between the two main cognitive processes that characterizes dragging practices [START_REF] Saada-Robert | La microgénèse de la rapresentation d'un problème[END_REF][START_REF] Arzarello | A cognitive analysis of dragging practices in Cabri environments[END_REF][START_REF] Olivero | Cabri-Géomètre as a mediator in the process of transition to proofs in open geometric situation[END_REF]: ascending and descending processes. We recognize ascending processes when students use the game in order to explore the situation and formulate a conjecture and descending processes when they use the game to check it. We have integrated this analytical tool with a new cognitive modality: the detached modality, in which students refer with words to the dynamic observed in the game, but they do not use it concretely.

The game of the two circles

The game-activity presented in this paper is based on the relationship between the distance between the centres of tangent circles and the sum/difference between their radii. Students play the game on the GeoGebra file shown in Figure 1. The GeoGebra window is divided in two parts: on the right there is the numerical window with sliders and variables, in the left the graphic window here is a graphic representation of the geometric objects. The game develops as follows: player B, the verifier, controls slider b, player C, the falsifier, controls slider c while player A, the referee, controls slider a and the hourglass. The goal of player B is to make e=d or e=f, the goal of player C is to make e≠d and e≠f. At the beginning of each match, the referee chooses the value of a and turns the hourglass over. Each time a player reaches his goal, the referee turns the hourglass over and the turn moves to the opponent. If the player cannot reach the aim within the time on his/her hands, he/she loses. The dynamic described is that of a semantical game played on the following statement: for every value of c there exists a value of b such that the circles are internally or externally tangent.

Each time that player B reaches his goal he produces an example of internally or externally tangent circles (look at Figure 2 a,b). Contrastingly, each time player C reaches his goal he produces an example of non-tangent circles (look at Figure 2 c,d,e). Since the interval of the sliders can take values from 0 to 10, players can produce also degenerate configurations (look at Figure 2 f,g). Player B can win also in this situation (look at Figure 2 h). Theoretically it is always possible for players to reach their goals. Therefor the outcome of the game is determined by the time limit.

After playing the game, students are required to answer to the following questions:

1. Which are the mutual positions between the two circles each time player B reaches his aim? 2. Which are the mutual positions between the two circles each time player C reaches his aim? 3. What do the sliders a, b and c represent? 4. What do the value of d, e and f represent?

The questionnaire is intended to help students shift their frame of reference from the game to the geometric theory. In particular, the first two questions are intended to change the focus of attention from the numerical values of variables d, e and f to the mutual positions between circles. In this way students discover the geometric invariants which characterizes player B's moves: each time the verifier reaches the goal the circles are tangent. Question number three is intended to link the values of the sliders to the length of the radii and the distance between the centres. Finally, question number four is intended to link the values of the parameters to the sum and difference of the radii. In this way students can discover another invariant which characterized player B's moves: each time the verifier reaches the goal the distance between the centres is equal to the sum or difference between the radii

Methodology and data collection

The study reported in this paper involves one classroom of 7 th grade Italian students. The game of the two circles is the first of a group of four game-activities related to the geometry topic of circles. Note that the properties on which the game are designed are not part of the classroom knowledge: the goal of the activity is to guide students in their discovery. Each activity lasts almost two hours: in the first hour and half students are divided into groups of three students and they play the game and answer a questionnaire using a computer or a tablet. In the last half an hour the teacher revisits students discoveries and systematizes the mathematical knowledge. The data for the analysis includes the transcript of students' dialogue and the GeoGebra diagrams explored during the game and the questionnaire. We videotaped two groups and the final class discussion but, for space reason, we will represent only one group's work.

Analysis

The videotaped group is composed by three students: Gu and Al are males, Bia is a female. They play the game on the computer. In the first match Bia is the referee, she chooses the value of a and she turns the hourglass. Gu is player C, the falsifier. He has to move slider c so that e≠d and e≠f. Al is player B, the verifier. He has to move slider b so that e=d or e=f. Figure 3 The match lasts short length time and it ends with the winning of the falsifier (Figure 3). In the last move the time ends before Al reaches his goal, hence Al loses even if, theoretically, he could have won. Al knows that he could have won if he had had more time, in fact he says "it should have been like this", making internally tangent circles. After Al demonstrates the winning configuration, Bia says "So B should always win". This sentence reveals the activation of the anticipatory thinking [START_REF] Harel | The development of mathematical induction as a proof scheme : A model for DNRbased instruction[END_REF]. After playing another match, students move to the first question.

Gu:

In order to reach the goal they have to touch each other in only one point.

Al:

On the other hand the answer to the question: 'In which mutual positions are the two circles when C reaches his/her goal?' is any position. They can touch each other in two points or nowhere.

Bia:

No, they always touch each other in exactly two points. (looking at an example of secant circles).

Gu:

They can touch each other in two points, but they can also not touch each other.

Bia: Ah… (moving c so that the circles do not intersect each other. Then she moves c back and forth for 30 seconds) Yes, that's right! Al and Gu approach the question in a different way from Bia. They are in detached modality, they rethink what has happened in the played-game and then they answer the question. Bia, instead, uses the game in order to investigate the situation, she is in descending modality: she is using the reflectedgame in order to check her schoolmates' claims. The group repeats this approach (detached versus descending) in answering the subsequent questions. When they get to the last one, the students do not agree with each other: according to Al and Bia, d and f are the radii of the circles, while Gu does not agree with them. The disagreement invokes the need of a justification.

Al:

Let me prove that it's the radius (taking the mouse)! They have to coincide perfectly… (moving the centre O' on the other circumference, see Figure 4e) Al: Don't you notice? Don't you see? (he moved c to 0, obtaining that e is equal to d and f, see Figure 4f). Point T just appeared and, putting this on zero. Do you notice? It's 2.9 2.9 2.9.

In order to refute Al's conjecture, Gu tries to explain the contradiction he noticed between the graphic and the numerical window. In detached modality, Gu explains why Al's claims creates a dynamic contradiction in the conversion [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] from the numerical to the graphic register. Al, instead, tries to provide evidence for explaining that what he claims is true. In this attempt, he uses the value of the parameter e (distance between the centres) in order to measure the length of the radius of one circle. His goal is to show that this value is equal to the value of d or f. If this process were to be applied on a generic example, it would have led to a contradiction, but since Al moves the value of the radius of the circle O' to 0, he produces a supporting example.

e) f) g) h)

Figure 4: Reflected-game

After exploring silently the situation, Bia who at first supports Al's claim, changes her mind.

Bia: Anyway he's right… If you put them like this (each centre belongs to the other circle, see Figure 4g), the radius is the same thing, isn't it? I mean, it's the same here and here (pointing at the two circles), but here they are different (pointing at the values of sliders d, f)… Then it must be another thing, do you get it?

Adopting Al's strategy, she uses the distance between the centres to create two circles with the same radii in the graphic window. She notices that the two parameters which are supposed to be the radii are not equal in the numerical window. In contrast to Gu, who creates a dynamic counterexample, Bia exhibits a static counterexample, but this one also fails to convince Al that he is mistaken. The discussion continues with Gu who repeats his dynamic example, Bia who produces other static counterexample and Al who moves back to his supporting special example and cannot understand why he is wrong. Finally, observing the value d=0 when circles are externally tangent (Figure 4h)

and with the same radii, Al formulate a new conjecture: d is the difference between the radii of the circles. This discovery allows him to unlock the situation and to explain his special example.

Discussion and conclusion

Game-activities can operationalize a functional approach to geometry. Within these activities, students deal with soft tangent circles, namely dynamic circles in which some constructive steps that make the circles robustly tangent (the tangency is preserved by dragging) are voluntarily not performed [START_REF] Healy | Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions[END_REF]. A constructive step creates a functional dependence between the geometric elements, which is hidden in the robust construction of the figure. In DGE game-activities, this functional dependence is made explicit through the verifier/falsifier's dialectic. More precisely, when the verifier has observed the invariant tangent configuration produced by his/her moves, he/she can create a cause-effect link between his/her goal and the invariant produced.

We describe the verifier's dragging as follows: 𝑏 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 → 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 which indicates that the verifier, by moving the slider b, can observe the invariant tangent configuration as the effect of making sliders values coincide. Once discovered the invariant, the verifier can accomplish the move with the goal of building tangent circles. In this case, he observes the coincidence of the values of the slider as the effect of making tangent circles. This time the verifier's dragging is described in this way: 𝑏 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 → 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑙𝑖𝑑𝑒𝑟. By switching the focus of attention of the move, the DGE game-activities, create a sort of frame, which helps students to appreciate the "if and only if" relationship between tangent circles and the fact that the distance between the centres is equal to the sum or absolute values of the difference of the radii. DGE game-activities enrich the exploration supporting the in-depth investigation of situations: the presence of the falsifier, who tries to create trouble to the verifier, exposes the verifier to different initial situations triggering the exploration of both standard and non-prototypical examples of tangent circles. In this way, the game-activities enlarge students' accessible example space [START_REF] Goldenberg | Shedding light on and with example spaces[END_REF]) associated with tangent circle configuration. This is a very important aspect for the construction of mathematical concepts: proposing students only standard configurations can be source of mathematical misconceptions.

Finally, the game tool enriches and supports students' arguing abilities and the coordination of numerical and graphic information. In order to communicate their claim, students activate a versatile use of the game: not only for formulating and checking conjectures but also for supporting their claim, confuting different opinions and explaining ones' point of view. The game assumes a fundamental role in promoting mathematical ways of reasoning. Al, for example, uses the game to show evidences of the truth of his claim, hence uses the game for constructing a supporting example, while Bia and Gu use it to show that he is wrong, hence for constructing counterexamples to what has been claimed by Al. In producing these arguments, students make conversion between numerical and graphic registers. Concluding, the game instruments help students not to assume the absolute truth of external opinions, but to establish a dialectic approach to them.

  Sliders a, b, and c control respectively the distance between the centres, the radius of the circle with centre O and the radius of the circle with centre O'. The variables d, e, f are respectively the absolute value of the difference between the radii (d=|b-c|), the distance between the centres and the sum of the radii (f=b+c). When students drag sliders b or c, they can observe the synchronic variation of the values of d and f and of the length of one circumference.

FigureFigure 2 :

 2 Figure 1: Game-activity

  Figure 3: First match

  not the radius… Because if you change the radius of a circle, you don't automatically change the radius of the other one! Both values [d and f] change! You should change just one [d or f] by moving it [b] by changing the length of one radius… you are not changing the other one!

Proceedings of CERME10