Chiara Sinigaglia
email: chi.sinigaglia@gmail.com

Michalis Averof

The multifaceted role of nerves in animal regeneration

published or not. The documents may come

The multifaceted role of nerves in animal regeneration

Introduction

The discovery that salamander limb regeneration depends on elements of the nervous system [1**] provided one of the first insights into the regulation of regenerative processes. Analogous influences of the nervous system were subsequently found in planarians, annelids, echinoderms, arthropods and many other animal groups, suggesting that nerve-dependency might represent a conserved feature of animal regeneration [2,3].

Research on the topic has focused on vertebrates, particularly on amphibians, which have become the paradigm for understanding the mechanisms of nerve-dependent regeneration (extensively reviewed in Ref. [4]). In other animals, experimental approaches have been largely limited to surgical and pharmacological manipulations, to demonstrate the active role of nerves in regeneration. Technical limitations have hampered our understanding of molecular and cellular mechanisms that underpin this phenomenon in most animals, thus limiting cross-species comparisons and a broad evolutionary understanding [5**,6*]. Recent discoveries on the shared molecular mechanisms governing the development and physiology of nervous systems in animals (e.g. [7].), and the ongoing efforts to apply -omics, functional genetics and imaging approaches to a broader array of species [8] are now paving the way for re-addressing long-standing questions in this field.

What is nerve-dependency?

The term 'nerve-dependency' has been employed to refer to diverse phenomena. In its broadest sense, nerve-dependency would encompass any case where the presence of a nerve is needed for the correct unfolding of the regenerative process. The nerves could provide spatial or temporal signals. They could act directly, or indirectly through other cell types. Their effects could be all-ornone or graded, permissive or instructive.

There are three broad ways, in which the nervous system has been shown to contribute to regeneration (Figure 1): a) Source of local signals: Nerve-dependency can be directly mediated through the local action of the nerve terminals, which release signaling molecules. In axolotl limbs, the growth factor FGF-2 produced by the nerve promotes the formation of the Apical Ectodermal Cap (AEC). The subsequent interactions between nerves and the AEC drive the regenerative process [9]. b) Support for other cell types: Nerves can also contribute by serving as a scaffold for other cell types with important roles, such as cells that provide essential signals or serve as progenitors during regeneration. These may be cells that are closely associated with nerves even in nonregenerative conditions, such as the Schwann cells and their precursors, which provide essential signals for regeneration in salamander limbs [10**,11] and in mouse digit tips [5**], or mesenchymal progenitors that contribute to bone formation during digit tip regeneration [12**]. Equally, nerves can support other cell types that are not usually present in nerves but become transiently associated with them during regeneration. An example is the granule cell, a crinoidspecific cell type, which migrate along nerves and release essential growth factors [13]. Such transiently associated migratory cells also include progenitor cells, such as annelid neoblasts migrating along the ventral nerve cord to the wound site [14], or the scattered cells described in some sipunculid species, which leave the ventral nerve cord to generate epidermis and nervous system at the wound site [15]. c) Source of systemic signals: The nervous system may also perform systemic functions, essential for regenerative success -for example by releasing neurohormones in the circulatory system. This mode of action has been suggested for caudal regeneration of several annelid worms, which is impaired by brain ablation even in the presence of nerves at the wound site [16]. While nerves are not implicated in many aspects of embryonic development, they are known to play a role in specific cases. In development, as in regeneration, nerves can serve as a source of local instructive signals, a support or scaffold for other cell types, or a source of neuroendocrine signals that control development, metamorphosis and growth in a systemic fashion (e.g. [17]).

Phylogenetic and methodological disparity

Nearly two centuries of research on nerve-dependent regeneration produced a vast amount of disparate data, encompassing most animal groups (Figure 2, and Appendix). Vertebrate appendages (reviewed in Refs. [18,19]), echinoderm appendages (reviewed in Ref. [20]) and annelid bodies (reviewed in Ref. [21]) were extensively studied. Other taxa or regenerative processes received less attention, but nevertheless featured clear cases of nerve-dependent regeneration. For example, the presence of the nervous system was shown to be essential for body regeneration in a nemertean worm [22] and a holothurian [23]. Most animal phyla include members with the ability to regenerate complex body parts (dark grey boxes). An influence of the nervous system during regeneration has been found in numerous instances (red boxes), while the molecular players involved in nerve-dependent regeneration (identified by pharmacological assays, immunological or genetic approaches) have been identified in some cases. Lack of evidence does not imply absence of nerve-dependent regeneration. n.d., not determined. Silhouettes adapted from PhyloPic.org, further information can be found in the Appendix.

The methodological approaches used to investigate the role of nerves have been very diverse, influenced by the history of technological developments and by the specific opportunities and constraints presented by each species. The purely surgical approaches employed in the early experiments, featuring burning needles to ablate the nervous system (e.g. classic study on a brittle star [24*]) or transplantation to re-combine regenerating tissues with nerves (as extensively applied to annelids, e.g. [25]), provided solid proof of the influence of the nervous system on regeneration, identifying nerve-dependent and nerve-independent phases [26**]. Later work focused on 'inducing' or 'trophic' factors mediating the effect of nerves. Biochemical analyses of body and nerve extracts identified candidate active substances, such as the organisines thought to be released by regenerating planarian brains [27], or the mitogenic glial growth factor detected in regenerating newt limbs [28]. Immunological and pharmacological experiments allowed to detect and to test the function of specific compounds, in vivo and on cell cultures, identifying for example the mitogenic function of transferrin in amphibian limb regeneration [29]. Gene-oriented approaches provided molecular insights into the interactions regulating nerve-dependency [30]. RNAi screens on planarians identified several genes that are expressed in the nervous system and promote regeneration, including NB42.2h and roboA [31*,32]. Differential expression analyses and a yeast-two hybrid screen allowed the identification of the Prod1 receptor and its ligand nAG, which mediate the interaction between Schwann cells and the AEC in regenerating salamander limbs [10*,33].

The multiple contributions of nerves

Forgotten studies, incomplete knowledge, and diversesometimes outdatedtechnical approaches represent a challenge for a thorough re-evaluation of nerve dependency in regeneration, particularly when searching for shared regulators. A systematic comparison across species offers nevertheless an interesting perspective over their multiple roles. Nerves can play an essential role at different stages of regeneration -wound healing, cell proliferation, cell differentiation, patterning and morphogenesis.

Wound closure does not appear to depend strictly on the presence of nerves, but mounting evidence suggests that the nervous system could play a facilitating role in this process. Reepithelialization at the amputation plane in the absence of nerves has been reported for several species, including annelid worms [34], sea stars [35], zebrafish [36]. However, in several vertebrates, including mammals [37] and birds [38], denervation delays cutaneous wound healing. In amputated lobster limbs the migration of ectodermal cells appears more pronounced in regions that lie closer to nerves [39].

Severed nerves could facilitate wound healing by helping to detect injuries and to mount an immune response. Cut nerve terminals are among the first tissues to react after amputation, by retracting from the wound site and undergoing degeneration; examples are found in cephalopod molluscs [40], brittle stars [41], decapod crustaceans [42] and phoronids [43]. Injured nerves also stimulate macrophage recruitment [44,45]. Macrophages are necessary for cell proliferation at the amputation site; their depletion blocks regeneration in both salamander limbs [46] and zebrafish fins [47].

An increase of cell proliferation is a common regenerative response of most studied systems (reviewed in Ref. [48]). Proliferating cells may be broadly distributed (e.g. in asteroids), or localized close to the amputation plane, in a proliferating mass called blastema (e.g. in salamanders, planarians and crinoids) [49,50].

Nervous systems have been shown to promote cell proliferation in numerous regenerating species, regardless of their regenerative strategies. Denervation experiments caused a reduction in the number of mitoses and prevented the formation of a histologically detectable blastema in regenerating newt limbs/tails [51], and in the ear of a regenerating mouse strain [START_REF] Buckley | Denervation affects regenerative responses in MRL/MpJ and repair in C57BL/6 ear wounds[END_REF]. The molecular mediators of this function have been characterized in few vertebrate systems. In salamander limbs [10*] nerves play an essential role in triggering/sustaining cell proliferation in the blastema by secreting nAG. This protein binds to a receptor, Prod1, identified in responding blastemal cells, and elegant experiments have shown that the effects of limb denervation can be rescued by providing exogenous nAG, even if nerve and muscle tissues could not regenerate normally [10**]. Cholinergic nerves promote cardiomyocyte proliferation during heart regeneration in both mouse and zebrafish [START_REF] Mahmoud | Nerves regulate cardiomyocyte proliferation and heart regeneration[END_REF].

In invertebrates, molecular evidence for a role of nerves in inducing cell proliferation is overall lacking. Some descriptive evidence would be consistent with such a role, such as the close proximity of nerves to proliferating cells (e.g. in crustaceans [42], enteropneusts [START_REF] Rychel | Anterior regeneration in the hemichordate Ptychodera flava[END_REF], echinoderms [41], and annelids [START_REF] Coulon | Relation entre l'innervation et l'activité prolifératrice des cellules blastematique au cours de la régénération de l'annélide polychete Owenia fusiformis: études ultrastructurale et autoradiographique[END_REF]) and the fact that cell proliferation starts concomitantly with the regrowth of nerves into the blastema (e.g. in cuttlefish [40]). Growth defects or delays in regeneration have been observed following denervation, particularly among regenerating arthropod limbs [56*,57].

The influence of nerves on the differentiation of progenitor cells and on growth is even less well understood. In hydra, nerve cells secrete peptides controlling the spatial and temporal patterns of differentiation [START_REF] Schaller | Neuronal control of development in hydra[END_REF]. In the sea star, regeneration is entirely nerve-dependent, and denervation blocks the differentiation of progenitor cells, and the growth of the regenerating arm [35].

Innervation can also influence patterning and polarity. Clonal cell lineage tracing in mice revealed morphological defects in regenerating finger tips upon denervation, indicating that nerves affect tissue patterning in both ectodermal and mesodermal derivatives [START_REF] Rinkevich | Clonal analysis reveals nerve-dependent and independent roles on mammalian hind limb tissue maintenance and regeneration[END_REF]. The nerve cord of annelids was shown to impart ventral identity to the regenerate (reviewed in Ref. [START_REF] Oviedo | Long-range neural and gap junction protein-mediated cues control polarity during planarian regeneration[END_REF]), while the central nervous system of planarians controls the polarity of regenerating body fragments by detecting the presence of cephalic structures [START_REF] Herbst | Über die Regeneration von antennenähnlichen Organen an Stelle von Augen[END_REF]. Furthermore, the identity of regenerating elements might also depend on their nerve supply: in several crustaceans, the presence of the optic ganglion appears to be required for regenerating an eye and removal of that ganglion leads to regenerating an antenna-like structure [START_REF] Bailey | The influence of the nervous system in the regeneration of Eisenia foetida savigny[END_REF].

Denervation commonly results in the formation of a stump that fails to regenerate, as classically shown after amputation of posterior body segments of annelids [START_REF] Géraudie | Nerve dependent macromolecular synthesis in the pectoral fin regenerate of the fish, Fundulus[END_REF], echinoderm arms [24*], salamander limbs and fins of teleost fish [START_REF] Schotté | Morphological effects of denervation and amputation of limbs in urodele larvae[END_REF]. In some instances, however, the removal of nerve induces a more dramatic response, where also tissues far from the wound degenerate. This phenomenon, observed initially in the denervated and amputated limbs of larval urodeles, has been termed the 'regression effect' [START_REF] Karczmar | The role of amputation and nerve resection in the regressing limbs of urodele larvae[END_REF][START_REF] Drescher | Regenerationsversuche am gehirn von Periplaneta Americana unter berucksichtigung von verhaltensunderung und neurosekretion[END_REF]. In the denervated antennae of cockroaches, the amputated stump degenerated over several molts, but could regenerate once it was re-innervated [67*].

Re-routing nerves to a wounded area can also have a converse gain-of-function effect, inducing the ectopic formation of a regenerating structure. This effect has been best studied in axolotl limbs (termed the Accessory Limb Model, [START_REF] Ruben | Delayed denervation and accessory limb formation in urodeles[END_REF][START_REF] Bely | Early events in annelid regeneration: a cellular perspective[END_REF]), and also described during anterior regeneration in annelids [START_REF] Nacu | FGF8 and SHH substitute for anterior-posterior tissue interactions to induce limb regeneration[END_REF].

Searching for common molecular mediators

The molecular regulation of nerve-dependent regeneration remains poorly understood, and most of the factors identified so far derive from vertebrate research (e.g. nAG and Prod1 [10**], FGFs [9,[START_REF] Makanae | Co-operative Bmp-and Fgf-signaling inputs convert skin wound healing to limb formation in urodele amphibians[END_REF], GGF2 [28], BMP [START_REF] Boilly | Nerve dependence: from regeneration to cancer[END_REF], transferrin [29], and Neuregulin [6*]reviewed in Refs. [19,21,[START_REF] Satoh | Nerve roles in blastema induction and pattern formation in limb regeneration[END_REF]).

Among the wide range of signaling molecules produced by nerves, neurotransmitters might represent the best candidates for conserved molecular mediators. Besides their well-known roles in neuronal communication, neurotransmitters have been implicated in development and shown to influence regeneration in several contexts [START_REF] Lauder | Neurotransmitters as morphogens[END_REF]. In planarians dopamine levels vary significantly during regeneration, and treatment with a dopamine antagonist reduces cell proliferation, delaying regeneration [75*]. Analogous results have been obtained in the regenerating arms of sea star [START_REF] Huet | Histofluorescence study and biochemical assay of catecholamines (Dopamine and Noradrenaline) during the course of arm-tip regeneration in the starfish, Asterina gibbosa (Echinodermata, Asteroidea)[END_REF] and in regenerating fragments of a sea anemone [START_REF] Lenicque | Effects of biogenic amines on the regeneration of small pieces of the pedal disc of the sea anemone Metridium senile (linnaeus)[END_REF]. Significant fluctuations in serotonin concentration during arm regeneration have been detected in crinoids [START_REF] Sugni | New evidence of serotonin involvement in the neurohumoral control of crinoid arm regeneration: effects of parachlorophenylanine and methiothepin[END_REF]. Drug treatments have shown that serotonin promotes regeneration in planarians and that this role is counterbalanced by acetylcholine signaling [START_REF] Lenicque | A mechanism of action of neurotransmitters on the regeneration of the planarian worm Dugesia tigrina. Role of acetylcholine as a negative feed-back[END_REF]. Similarly, noradrenaline has been found to promote regeneration in planarians [75*] and annelids [START_REF] Chapron | Etude comparée de la nature et du mode d'action des substances neurotrophiques intervenant dans la régénération des oligochétes et des amphibiens[END_REF] and has been detected in the regenerating limbs of newt, where its levels correlate with cAMP activity [START_REF] Taban | Cyclic AMP and noradrenaline sensitivity fluctuations in regenerating newt tissue[END_REF].

Neuropeptides are signaling molecules that mediate local interactions between neurons or act systemically as hormones [START_REF] Elphick | Evolution of neuropeptide signalling systems[END_REF][START_REF] Jékely | The long and the short of ita perspective on peptidergic regulation of circuits and behaviour[END_REF]. Substance P and substance K (Neurokinin A), two neuropeptides of the tachykinin family, have been shown to have a pro-mitogenic effect on diverse tissues [START_REF] Zachary | A role for neuropeptides in the control of cell proliferation[END_REF]. Treatment with substance P and K promotes neoblast proliferation in regenerating planarians [START_REF] Baguna | Effects of activators and antagonists of the neuropeptides substance P and substance K on cell proliferation in planarians[END_REF]. Substance P was also detected in the proliferating blastema of newt limbs, where it similarly stimulates regeneration [START_REF] Smith | Nerve extracts and substance P activate the phosphatidylinositol signaling pathway and mitogenesis in newt forelimb regenerates[END_REF]. Neuropeptide F (NPF) stimulates mitotic activity of neoblasts and promotes pharyngeal regeneration in planarians [START_REF] Kreshchenko | Functions of flatworm neuropeptides NPF, GYIRF and FMRF in course of pharyngeal regeneration of anterior body fragments of planarian, Girardia tigrina[END_REF]. Interestingly, a pro-mitogenic action of NPF was recently shown in the germ stem cells niche of Drosophila [START_REF] Ameku | Midgut-derived neuropeptide F controls germline stem cell proliferation in a mating-dependent manner[END_REF]. Hydra polyps regeneration depends on the Head Activator (HA) neuro-peptide, which promotes proliferation and differentiation of nerve and epithelial cells, via the cAMP pathway [START_REF] Schaller | Neuronal control of development in hydra[END_REF][START_REF] Schaller | Isolation and amino acid sequence of a morphogenetic peptide from hydra[END_REF]. In mammals, HA promotes proliferation of nervous system or neuroendocrine-derived cells, during development [START_REF] Rezgaoui | The neuropeptide head activator is a high-affinity ligand for the orphan G-protein-coupled receptor GPR37[END_REF]. Neuropeptides from cutaneous nerves contribute to wound healing in vertebrates: gastrinreleasing peptide (GRP, a homologue of amphibian bombesin) promotes wound healing in mammals by stimulating growth and spreading of keratinocytes [START_REF] Baroni | Bombesin: a possible role in wound repair[END_REF].

Nerve-dependency and the evolution of animal regeneration

A widespread requirement for the presence of nerves emerges from the comparison of animal regenerative processes, suggesting that the nervous system could play a conserved role. Given the ongoing efforts to identify any unifying principles in the unevenly distributed regenerative capacities of animals, finding a shared role for the nervous system would have profound consequences for our understanding of the evolution of regenerative properties [92**].

The evidence reviewed here, however, highlights the diversity of the various processes classified as 'nerve-dependent regeneration'. The function of nerves can be mediated locally by the axons themselves, by cells variably associated with the nervous system, or systemically by circulating signals. Nerves can intervene at different stages of the regenerative process, with the control of proliferation being perhaps the most broadly shared feature. At present, it is uncertain which aspects could be considered homologous and compared across widely divergent groups, in terms of the mode of action, the cell types involved or the molecular mediators. Within vertebrates, the scaffolding/support that nerves provide for factor-secreting glial cells could be considered as a shared feature, but the molecular regulators only partially overlap between salamander limbs (the role of nAG is not well conserved), mouse digits [5**] or teleost fins [36]. The molecular mediators might evolve rapidly, even if the overall process is conserved.

The lack of comparable molecular data across animals represents an important obstacle for the emergence of a unified view. Cephalopod molluscs, despite their excellent regenerative abilities [START_REF] Imperadore | Cephalopod tissue regeneration: consolidating over a century of knowledge[END_REF] and the extensive knowledge about the physiology of their nervous systems [START_REF] Katz | Phylogenetic plasticity in the evolution of molluscan neural circuits[END_REF], appear to have been largely overlooked with regard to nerve-dependent regeneration. Similarly, no relevant molecular data are available for cephalochordates or ctenophores, two groups with high regenerative capabilities and relevant phylogenetic positions [START_REF] Somorjai | Vertebrate-like regeneration in the invertebrate chordate amphioxus[END_REF][START_REF] Martindale | The onset of regenerative properties in ctenophores[END_REF]. The discovery of neuropeptide signaling in placozoans, a non-bilaterian group lacking a nervous system [START_REF] Elphick | Evolution of neuropeptide signalling systems[END_REF][START_REF] Varoqueaux | High cell diversity and complex peptidergic signaling underlie placozoan behavior[END_REF], opens the intriguing possibility that neuropeptides might be involved in regeneration even in the absence of a nervous system.

Tools are now becoming available to tackle the evolution of nerve dependent regeneration through a broader sampling of animal groups. Achieving denervation by means of precise laser ablation or inducible genetic cell ablation systems, instead of invasive surgery, now allow to spatially and temporally control the interactions between nerves and target cells. Transgenic reporters and in vivo cell tracking approaches provide more powerful/specific ways to assess the effects that denervation has on regeneration, for example by uncoupling the dynamics of axons and nerveassociated cells. These approaches could be applicable and particularly informative in arthropods, where parallels with vertebrate regeneration such as the role of satellite-like cells in regenerating muscles have already been established [START_REF] Konstantinides | A common cellular basis for muscle regeneration in arthropods and vertebrates[END_REF]. Classical ablation studies in cockroaches and crustaceans have thus far only revealed subtle effects of removing nerves during regeneration, possibly due to the fast regrowth of axons into the stump [START_REF] Bodenstein | Contributions to the problem of regeneration in insects[END_REF]. In vivo imaging of regeneration using transgenic reporters, in the crustacean Parhyale [100], would allow to effectively monitor the denervation process and to dissect the cellular events that follow.

Nerve-dependent regeneration is a complex process and understanding its evolution will be challenging. The phenomenon is so widespread that it could be ancestral, despite the high variability observed between and within animal groups. Alternatively, there may be no unifying principle; nerves may have been co-opted into the regenerative process multiple times with different roles. Investigating the evolution of nerve-dependent regeneration with a common approach in a broader sample of animals will allow for a better understanding of the evolution of regenerative developmental mechanisms.

Conflicts of interest statement

Nothing declared. This seminal study opened the field of nerve-dependent regeneration research, even before the development of a cell theory. Todd studied regeneration -termed 'reproduction'of salamander limbs and tails, and demonstrated the influences of innervation on the process. He correctly identified the critical windows of sensitivity by performing denervation experiments at different stages of regeneration. The authors identified a role for nerve-associated Schwann cell precursors (SCPs) in the regeneration of mouse digit tips, thus showing that glial cells are also required for regeneration in mammals. SCPs dedifferentiate and secrete oncostatin M and platelet-derived growth factor AA, which promote blastema growth and can rescue defects of denervated digits. Perturbation of SCPs signaling causes morphogenesis defects, in both mesodermal and ectodermal derivatives. This study identified nerve secreted Neuregulin-1 (NRG1) as an essential mediator of the interaction between nerves and Schwann cells during regeneration in axolotl. NRG1 localizes to the wound epithelium before blastema formation, and is later expressed in proliferating blastemal cells, together with its receptor ErbB2. Perturbation of NRG1 suppresses regeneration, while additional expression rescues denervated limbs.

Figure 1 :

 1 Figure 1: Modes of action of the nervous system during regeneration. The nervous system has been shown to contribute to regenerative processes in different ways, across animal groups. (a) Nerves can directly act as a source of local signals. (b) They can act as a support for other cell types, which provide signals and/or act as progenitors in the regenerating tissues. (c) Systemic signals originating from the central nervous system can also influence the regenerative process.

Figure 2 :

 2 Figure2: Distribution of nerve-dependent regeneration in animals. Most animal phyla include members with the ability to regenerate complex body parts (dark grey boxes). An influence of the nervous system during regeneration has been found in numerous instances (red boxes), while the molecular players involved in nerve-dependent regeneration (identified by pharmacological assays, immunological or genetic approaches) have been identified in some cases. Lack of evidence does not imply absence of nerve-dependent regeneration. n.d., not determined. Silhouettes adapted from PhyloPic.org, further information can be found in the Appendix.

2 .

 2 Dinsmore CE: A History of Regeneration Research. Milestones in the Evolution of a Science. Cambridge University Press; 1991.3. Farkas JE, Monaghan JR: A brief history of the study of nerve dependent regeneration. Neurogenesis 2017, 4:e1302216. 4. Kumar A, Brockes JP: Nerve dependence in tissue, organ, and appendage regeneration. Trends Neurosci 2012, 35:691-699. 5. ** Johnston AP, Yuzwa SA, Carr MJ, Mahmud N, Storer MA, Krause MP, Jones K, Paul S, Kaplan DR, Miller FD: Dedifferentiated Schwann cell precursors secreting paracrine factors are required for regeneration of the mammalian digit tip. Cell Stem Cell 2016, 19:433-448.

6 .

 6 * Farkas JE, Freitas PD, Bryant DM, Whited JL, Monaghan JR: Neuregulin-1 signaling is essential for nerve-dependent axolotl limb regeneration. Development 2016, 143:2724-2731.

7 .

 7 Kelava I, Rentzsch F, Technau U: Evolution of eumetazoan nervous systems: insights from cnidarians. Philos Trans R Soc B Biol Sci 2015, 370:20150065. 8. Mokalled MH, Poss KD: A regeneration toolkit. Dev Cell 2018, 47:267-280. 9. Mullen LM, Bryant SV, Torok MA, Blumberg B, Gardiner DM: Nerve dependency of regeneration: the role of distal-less and FGF signaling in amphibian limb regeneration. Development 1996, 122:3487-3497. 10. ** Kumar A, Godwin JW, Gates PB, Garza-Garcia AA, Brockes JP: Molecular basis for the nerve dependence of limb regeneration in an adult vertebrate. Science 2007, 318:772-777.

This study identified nAG, a protein secreted by Schwann cell, which controls proliferation of blastema cells during newt limb regeneration. A yeast-two-hybrid screen identified nAG as the ligand for the surface receptor Prod1, expressed by blastema cells. Electroporation of nAG into denervated blastemas could rescue regeneration. 11. Kumar A, Delgado JP, Gates PB, Neville G, Forge A, Brockes JP: The aneurogenic limb identifies developmental cell interactions underlying vertebrate limb regeneration. Proc Natl Acad Sci U S A 2011, 108:13588-13593.

This study provided early molecular genetic evidence of nerve-dependency among invertebrates. It showed that defective nervous system in the planaria Schmidtea mediterranea impairs pattern formation during regeneration, leading to the development of ectopic structures. The nervous system was perturbed by RNAi knockdown of the axon-guidance molecule roundabout (Smed-roboA).

Acknowledgements

We thank Igor Adameyko for comments on the manuscript. Our work is supported by the European Research Council, under the European Union Horizon 2020 program, grant ERC-2015-AdG #694918. We apologize to all the authors whose work could not be cited due to space limitations.

References and recommended reading

Papers of particular interest, published within the period of review, have been highlighted as: * of special interest, ** of outstanding interest 1. ** Todd TJ: On the process of reproduction of the members of the aquatic salamander. Quart J Sci Arts Lib 1823, 16:84-86.

Appendix Supplementary Table. References related to Figure 2.

Nerve-dependent regeneration

References in main text

Further references

Ctenophora 96

Porifera


Laurent Foulloy

Eric Benoit

Building a Class of Fuzzy Equivalence Relations

Keywords: Fuzzy equivalence relations, approximate equalities, strict t-norms, reflexivity, fuzzy meanings and descriptions, fuzzy partitions

, the comparison of colored objects and comfort situations are proposed.

Introduction

In many applications, especially in fuzzy control, triangular or trapezoidal membership functions are used.

Moreover, these membership functions define a fuzzy partition in the sense of Ruspini [START_REF] Ruspini | A new approach to clustering[END_REF] as shown in figure 1. It is often recognized that this type of membership functions are simple to handle and to compute with. Only a

x A 1 A 2 A 3 A 4 A 5 A 6
*Corresponding author. Tel.: +(33) 450 096 549; fax: +(33) 450 096 559.

E-mail address: Laurent.Foulloy@univ-savoie.fr few papers have tried to bring sound motivations for what can be considered as the simplest possible form for membership functions. Pedrycz proposed two main interpretations: one is based on equalization, like entropy equalization, while the second one relies on the error-free reconstruction of a fuzzified signal by means of a defuzzification interface [START_REF] Pedricz | Why triangular membership functions?[END_REF], [START_REF] Pedrycz | Fuzzy equalization in construction of fuzzy sets[END_REF]. In [START_REF] Soto | Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator[END_REF], de Soto and Recasens, showed that fuzzy partitions based on triangular fuzzy numbers are obtained as compatible partitions associated with indistinguishability operators associated with the Lukasiewicz triangular norm. Dubois et al. also showed that triangular fuzzy numbers can be interpreted as the optimal probability-to-possibility transformation of a uniform probability distribution in a bounded interval [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF].

It is also quite conventional to use sum-product inference in fuzzy systems, like in Mizumoto fuzzy controllers for instance. It is worth noticing that the sum operator is not a t-conorm. However, when a triangular based fuzzy partition is used, the bounded sum, that is Lukasiewicz t-conorm, reduces to the sum operator and the output of the fuzzy system also satisfies Ruspini's condition on the cartesian product [START_REF] Benoit | Fuzzy sensor aggregation: application to comfort measurement[END_REF].

In this paper, we introduce a class of fuzzy equivalence relations whose reflexivity condition generalises Ruspini's condition. Compatible fuzzy partitions enable to obtain various types of membership functions, including the triangular ones. In section 2, basic definitions are provided. In section 3, it is shown in which conditions the class of relations which is studied in this paper becomes fuzzy equivalence relations. General properties are given, including the comparison between relations, their transitivity and the construction of fuzzy equivalence relations on cartesian products. A short discussion concludes this section. The links between fuzzy partitions and a linguistic approach are presented in section 4. Finally, several examples are proposed in section 5.

The first one concerns the comparison between colored objects. The second one deals with the fairy-tale example proposed by De Cock and Kerre [START_REF] Cock | On (un)suitable fuzzy relations to model approximate equality[END_REF]. Then, fairy-tale characters with colored clothes are used to illustrate an equivalence relation on a cartesian product. Finally, the analysis of comfort situations is proposed to emphasize the construction of fuzzy partitions associated with fuzzy equivalence relations.

Basic definitions

Definition 1: Let E be a set of vectors in [0, 1] n . Let T be a t-norm. Let be the mapping from to the positive real line, defined from the pair (E, T), as :

(1)

R T E E × a b , ( ) ∀ E 2 ∈ R T , a b , ( ) T a i b i , (
).

v) a T # -transitive fuzzy relation on E if it is a fuzzy relation on E and vi) a T # -equivalence on E if it is a fuzzy relation on E, reflexive, symmetric and T # -transitive.

Definition 3: The mapping , defined by , is called the diagonal section of the tnorm T [START_REF] Klement | Triangular Norms[END_REF].

Remark 1: Let be the t-norm defined by . The reflexivity condition for the mapping becomes:

(

This condition corresponds to Ruspini's definition of a fuzzy partition [START_REF] Ruspini | A new approach to clustering[END_REF].

Definition 4: Let be a finite family of fuzzy sets of X, such that , and φ be a [0, 1]automorphism. The family P is a φ-partition on X [START_REF] Baets | Fuzzy Partitions and their Entropy[END_REF], if it verifies:

. Remark 2: Thus, the reflexivity condition of the relations can be related to De Baets's and Mesiar's definition of φ-partitions, using as the [0, 1]-automorphism with T a strict t-norm. Let us recall that a t-norm

T is strict if it is continuous and strictly monotone, i.e. T(u, v) < T(u, w) whenever u > 0 and v < w (see [START_REF] Klement | Triangular Norms[END_REF] for a large covering on t-norms).

R T a E R , ∈ ∀ T a a , ( ) 1. = a b , ( ) ∀ E 2 ∈ R T a b , ( ) , 1. ≤ a b , ( ) ∀ E 2 ∈ R T a b , ( ) , R T b a , ( ). = a b c , , ( ) ∀ E 3 T # R T a b , ( ) R T b c , ( ) , ( ) R T a c , ( ). ≤ , ∈ φ: 0 1 , [ ] 0 1 , [ ] → φ u ( ) T u u , ( ) = T M : 0 1 , [ ] 2 0 1 , [ ] → T M u v , ( ) min u v , ( ) = R T M a E ∈ R T M a a , ( ) φ a i ( ) i 1 = n ∑ = , ∀
min a i a i , (

)

i 1 = n ∑ a i i 1 = n ∑ 1. = = = P A i { } i I ∈ = i I A i ∅ ≠ , ∈ ∀ x X φ A i x ( ) ( ) i I ∈ ∑ , ∈ ∀ 1 = R T φ u ( ) T u u , ( ) =

Properties of R T mappings

Generalities

Let T be a strict t-norm. It is well-known that strict t-norms are isomorphic to the product, that is there exists a strictly increasing bijection , such that, for all :

(3) Thus, we have for all :

(4)

As the bijection θ is strictly increasing, the function φ is also a strictly increasing bijection. The inverse function is defined, for all , by: . ( 5)

Remark 3: The bijection φ can also be defined from the additive generator of the strict t-norm T. If T is a strict tnorm, it has an additive generator which is a strictly continuous decreasing function such that t(0) = ∞, t(1) = 0 and:

Therefore, it can easily be deduced, for all : ,

.

Remark 4: It is always simple to build a set E of vectors in [0, 1] n such that the mapping is reflexive on E.

Equation ( 5) provides a simple means to define a set F such that, given a strict t-norm T with a diagonal section φ, the mapping is reflexive on F. Indeed, it can be defined as follows:

Obviously, we have:

θ: 0 1 , [ ] 0 1 , [ ] → u v , ( ) 0 1 , [ ] 2 ∈ T u v , ( ) θ 1 -θ u ( ) θ v ( ) ⋅ ( ) . = u 0 1 , [ ] ∈ φ u ( ) T u u , ( ) θ 1 -θ 2 u ( ) ( ). = = φ 1 - u 0 1 , [ ] ∈ φ 1 -u ( ) θ 1 - θ u ( ) ( ) = t: 0 1 , [ ] 0 ∞ , [ ] → u v , ( ) 0 1 , [ ] 2 T u v , ( ) , ∈ ∀ t 1 -t u ( ) t v ( ) + ( ) . = u 0 1 , [ ] ∈ φ u ( ) t 1 -2t u ( ) ( ) = φ 1 -u ( ) t 1 -t u ( ) 2 --------- ⎝ ⎠ ⎛ ⎞ = R T M R T F b 0 1 , [ ] n ∈ a ∃ E such that b ∈ i φ 1 -a i ( ), for all i 1, … n , { } ∈ = ; { } . = (10)
This property will be used in section 4 to define fuzzy partitions on ℜ.

F ∈ R T b b , ( ) φ b i ( ) i 1 = n ∑ = , ∀ φ φ 1 -a i ( ) ( ) i 1 = n ∑ a i i 1 = n ∑ 1. = = = R T M R T M T P : 0 1 , [ ] 2 0 1 , [ ] → R T P R T P T u v , ( ) φ , ( ) (
).

= =

Proposition 2: Let T be a strict t-norm. Let E be a set of vectors in [0, 1] n . Let be the mapping reflexive on E.

If the additive generator t of the t-norm T is strictly convex, then the following are equivalent for all :

(i) ,

(ii) , and, therefore, the relation is a reflexive fuzzy relation.

Proof. (ii) implies (i) from the fuzzy reflexivity condition of .

To prove that (i) implies (ii), let us remark that if the additive generator t of the t-norm T is strictly convex then, as it is decreasing by definition, its inverse t -1 is also strictly convex. Therefore, for all : [START_REF] Cook | Why T-equivalence relations do not resolve the Poincaré paradox, and related issues[END_REF] Due to the fuzzy reflexivity condition of , we have and , leading to: [START_REF] Soto | Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator[END_REF] Remark 5: This property is satisfied for the following families because their additive generators are strictly convex for the given parameter ranges:

• of Frank t-norms when λ ∈ ]0, ∞[, • of Schweizer-Sklar t-norms when λ ∈ ]-∞, 1[, • of Aczél-Alsina t-norms when λ ∈ [1, ∞[, • of Dombi t-norms when λ ∈ [1, ∞[, • of Hamacher t-norms when λ ∈ [0, 2[.
The family of Frank t-norms is continuous with respect to the parameter λ and we have where

T M R T a b , ( ) E 2 ∈ i 1, … n , { } a i b i ≠ , ∈ ∃ R T a b , ( ) 1 < R T R T a i b i ≠ 2t 1 -t a i ( ) t b i ( ) + ( ) t 1 -2t a i ( ) ( ) t 1 -2t b i ( ) ( ) + < 2T a i b i , ( ) T a i a i , ( ) T + b i b i , ( ). < ⇔ R T R T a a , ( ) 1 = R T b b , ( ) 1 =

2

T a i b i , (

) (  ) (  )

i 1 = n ∑ T a i a i ,
i 1 = n ∑ T b i b i ,
i 1 = n ∑ + < 2R T a b , ( ) R T a a , ( ) R T b b , ( ) + < ⇔ R T a b , ( ) 1. < ⇔ T λ F T λ SS T λ AA T λ D T λ H T 0 F T M =
is the minimum. Although T M is not a strict t-norm, proposition 2 holds true for this t-norm (Let us note that the proof without using the continuity of a family of t-norms is quite obvious). It can also easily be shown that all tnorms built as the ordinal sum [START_REF] Klement | Triangular Norms[END_REF] of strict t-norms with convex additive generators also satisfy the equivalence in proposition 2.

Comparison of R T fuzzy relations

In this section, after a general proposition, it will be shown that fuzzy relations can be compared with relations. Then, given two strict t-norms T 1 and T 2 , the condition under which the relation can be compared with is given. ------------------------------

R T R T M R T 1 R T 2 u v , ( ) 0 1 , [ ] 2 ∈ u′ φ 1 -u ( ) = v′ φ 1 -v ( ) = T u′ v′ , ( ) t 1 -t u ( ) t v ( ) + 2 ------------------------- ⎝ ⎠ ⎛ ⎞ . = T u′ v′ , ( ) t 1 -t u′ ( ) t v′ ( ) + ( ) = φ 1 -u ( ) t 1 -t u ( ) 2 --------- ⎝ ⎠ ⎛ ⎞ = T u′ v′ , ( ) t 1 -t t 1 -t u ( ) 2 --------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ t t 1 -t v ( ) 2 --------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ + ⎝ ⎠ ⎛ ⎞ t 1 -t u ( ) t v ( ) + 2 ------------------------- ⎝ ⎠ ⎛ ⎞ . = = φ R T a b , ( ) E 2 ∈ R T a b , ( ) R T M a′ b′ , ( ) ≥ a′ i φ a i ( ) = b′ i φ b i ( ) = i 1, … n , { } ∈ a′ i b′ i ≤ a′ i t 1 -t a′ i ( ) t b′ i ( ) + 2 -
- ⎝ ⎠ ⎛ ⎞ b′ i ≤ ≤ min a′ i b′ i , (
) ------------------------------

t 1 -t a′ i ( ) t b′ i ( ) + 2 -
- ⎝ ⎠ ⎛ ⎞ ≤ u φ a i ( ) = v φ b i ( ) =
Because t 2 is bijective, there exists such that and . Replacing in equation ( 16) leads to:

. (

Composing by and taking into account that is strictly decreasing gives:

Now, using proposition 3 with and , it comes:

. ( 19
)
Finally, since [START_REF] Janis | Resemblance in a nearness[END_REF] holds for all , we get:

and, therefore, .

Corollary 1: If is concave then and . ---------------------------------------------------- ---------------------------- -----------------------------

min a′ i b′ i , ( ) T a i b i , ( ) ≤ i 1, … n , { } ∈ R T a b , ( ) R T M a′ b′ , ( ) ≥ φ 1 φ 2 R T 1 R T 2 t 1 o t 2 1 - a b , ( ) E 2 ∈ R T 1 a b , ( ) R T 2 a′ b′ , ( ) ≥ a′ b′ , ( )
F 2 ∈ a′ i φ 2 1 -φ 1 a i ( ) ( ) = b′ i φ 2 1 -φ 1 b i ( ) ( ) = i 1, … n , { } ∈ t 1 o t 2 1 - x y , ( ) 0 ∞ , [ ] 2 ∈ t 1 o t 2 1 -x y + 2 ----------- ⎝ ⎠ ⎛ ⎞ t 1 o t 2 1 -x ( ) t 1 o t 2 1 -y ( ) + 2 -
≥ u v , ( ) 0 1 , [ ] 2 ∈ x t 2 u ( ) = y t 2 v ( ) = t 1 o t 2 1 -t 2 u ( ) t 2 v ( ) + 2 -
- ⎝ ⎠ ⎛ ⎞ t 1 u ( ) t 1 v ( ) + 2 -
≥ t 1 1 - t 1 1 - t 2 1 -t 2 u ( ) t 2 v ( ) + 2 ------------------------------ ⎝ ⎠ ⎛ ⎞ t 1 1 -t 1 u ( ) t 1 v ( ) + 2 ------------------------------ ⎝ ⎠ ⎛ ⎞ . ≤ u φ 1 a i ( ) φ 2 a′ i ( ) = = v φ 1 b i ( ) φ 2 b′ i ( ) = = T 2 a′ i b′ i , (
)

T 1 a i b i , ( ) ≤ i 1, … n , ( ) ∈ T 1 a i b i , (
)

i 1 = n ∑ T 2 a′ i b′ i , (
)

i 1 = n ∑ ≥ R T 1 a b , ( ) R T 2 a′ b′ , ( ) ≥ t 1 o t 2 1 - T 1 T 2 ≤ R T 2 a′ b′ , ( ) R T 1 a b , ( ) ≤
Proof. From the work of Schweizer and Sklar, it is known that, for continuous Archimedian t-norms, is subadditive and are equivalent [START_REF] Klement | Triangular Norms[END_REF]. As a corollary, if is concave then it is subadditive [START_REF] Klement | Triangular Norms[END_REF]. Thus, if is concave then and, according to proposition 5,

Transitivity of R T fuzzy relations

In this section, we first demonstrate that all fuzzy relations are T D -transitive, where

Three cases must be considered: i) and . In this case, according to proposition 2 and the reflexivity of , we have , therefore equation ( 21) holds true.

ii) or . According to the symmetry of T D , we will only consider the case where . The left hand side part of equation ( 21) is equal to . According to proposition 2 and the -------------------------------

t 1 o t 2 1 - T 1 T 2 ≤ t 1 o t 2 1 - t 1 o t 2 1 - T 1 T 2 ≤ R T 2 a′ b′ , ( ) R T 1 a b , ( ). ≤ R T R T M min u v , ( ) u v u v - - + 2 -
= R T P R T R T T D : 0 1 , [ ] 2 0 1 , [ ] → R T a b c , , ( ) ∀ E 3 T D R T a b , ( ) R T b c , ( ) , ( ) R T a c , ( ). ≤ , ∈ R T a b , ( ) 1 = R T b c , ( ) 1 = R T a b c = = R T a b , ( ) 1 = R T b c , ( ) 1 = R T a b , ( ) 1 = R T b c , ( )
reflexivity of , we have . Thus, equation ( 21) holds true.

iii) and . In this case equation ( 21) is always satisfied since the left hand side part is equal to 0. Proof. We have to show:

. ( 2 2 )
Let us replace by its definition:

If , Eq. ( 23) always holds true.

In the other case, the reflexivity of the relation gives which can be replaced in Eq. ( 23) leading to:

Using the distributivity of the addition with respect to the minimum, we have: [START_REF] Ruspini | A new approach to clustering[END_REF] which is always true.

R T a b = R T a b , ( ) 1 ≠ R T b c , ( ) 1 ≠ R T R T M R T M T L : 0 1 , [ ] 2 0 1 , [ ] → T L u v , ( ) max u v 1 - + 0 , ( ) = a b c , , ( ) ∀ E 3 max R T M a b , ( ) R T M b c , ( ) 1 0 , - + ( ) , ∈ R T M a c , ( ) ≤ R T M max T M a i b i , (
)

T M b i c i , ( ) + ( ) i 1 = n ∑ 1 -0 , ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ T M a i c i , (
).

i 1 = n ∑ ≤ T M a i b i , (
)

T M b i c i , ( ) + ( ) i 1 = n ∑ 1 - 0 ≤ b i i 1 = n ∑ 1 = T M a i b i , (
)

T M b i c i , ( ) + ( ) i 1 = n ∑ b i T M a i c i , ( ) + ( ) i 1 = n ∑ . ≤ T M a i b i , (
)

T M b i c i , ( ) + ( ) i 1 = n ∑ T M a i b i + c i b i + , ( ) i 1 = n ∑ , ≤ Proposition
If , Eq. ( 26) always holds true.

In the other case, according to equation ( 1), is the dot product of the vectors a and b. Thus, we have to show:

Let us note that the vectors in E are unit vectors due to the reflexivity condition of the fuzzy relation . Let us denote respectively the three vectors a, b, and c. The norm of the vector is given by: (

According to the triangular inequality, we have and therefore [START_REF] Valverde | On the Structure of F-indistinguishability Operators[END_REF]

R T P R T P T 0.5 Y T λ Y : 0 1 , [ ] 2 0 1 , [ ] → T D u v , ( ) if λ 0, = , max 1 1 u - ( ) λ 1 v - ( ) λ + ( ) 1 λ --- - 0 , ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ otherwise. , ⎩ ⎪ ⎨ ⎪ ⎧ a b c , , ( ) ∀ E 3 max 1 1 R T P a b , ( ) - 1 R T P b c , ( ) - + ( ) 2 - 0 , ( ) , ∈ R T P a c , ( ) ≤ 1 R T P a b , ( ) - 1 R T P b c , ( ) - + ( ) 2 1 > R T P a b , ( )
1 a b ⋅ - 1 b c ⋅ - + 1 a c ⋅ - . ≥ R T P OA OB OC , , AB AB b i a i - ( ) 2 i 1 = n ∑ a i 2 b i 2 2a i b i - + ( ) i 1 = n ∑ 2 1 a b ⋅ - . = = = AB BC + AC ≥ T L T λ Y a ]0 1[ , ∈ n ℵ ∈ a T n ( ) T # a a … a , , , ( ) 0 = = n times T D T 0 Y = ϕ: 0 1 , [ ] 0 1 , [ ] → : (29)
This remark opens an interesting question. Given a set E of vectors in [0, 1] n and a t-norm T with a strictly convex additive generator, can we find the greatest T # nilpotent t-norm such that the relation reflexive on E is a T # equivalence on E ?

As a first step towards an answer, figure 2 represents the highest λ, obtained by numerical computations, for which the relation associated with the family of Frank t-norms is -transitive on E. The x-axis is the base 10 logarithm of Frank's t-norm parameter α.

Fig. 2 -transitivity of the family of Frank t-norms.

Fuzzy equivalence on E 1 ×E 2

Equivalence relations are closely related to pseudo-metrics. In particular, if Q is a T # -equivalence on E then

d Q : E 2 →[0, ∞] defined by d Q = t #
oQ, with t # the additive generator of T # , is a pseudo-metric on E [START_REF] Baets | T-partitions, T-equivalences and pseudo-metrics[END_REF]. In this section, we show how to build a T # -equivalence on E 1 ×E 2 , given two T # -equivalences respectively on E 1 and E 2 .

Thus, it makes it possible to keep the same pseudo-metric on the cartesian product.

Proposition 9: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let E 1 and E 2 be two sets of vectors respectively in [0, 1] n and [0, 1] m . Let and be the two T # -equivalences respectively on E 1 and E 2 . Let be the strict t-norm defined by: . Then, for all and , the mapping defined by:

u v , ( ) 0 1 , [ ] 2 ∈ T # u v , ( ) ϕ 1 -T L ϕ u ( ) ϕ v ( ) , ( ) ( ) ϕ 1 -max ϕ u ( ) ϕ v ( ) 1 - + 0 , ( ) ( ) . = = R T R T α F T λ Y 10 - 6 10
T λ Y R T 1 R T 2 T*: 0 1 , [ ] 2 0 1 , [ ] → T* u v , ( ) φ 1 -φ u ( ) φ v ( ) ⋅ ( ) = a a′ , ( ) E 1 2 ∈ b b′ , ( ) E 2 2 ∈ R T , is a T # -equivalence on the cartesian product E=E 1 ×E 2 .
Proof. First of all, let us note that T* is a strict t-norm because, as shown in section 3, φ is a strictly increasing bijection from [0, 1] to [0, 1], which can therefore be used as a multiplicative generator. Now, to prove the proposition, as the additive generator of the t-norm T is strictly convex, it is sufficient to prove that the fuzzy relation is reflexive on E. The reflexivity of the relation is given, for all (a, b) ∈ E 2 , by: (

Replacing T* by its definition, then using the distributivity of the sum with respect to the product and, finally, the reflexivity of the relations and respectively on E 1 and E 2 , we have:

Discussion

It is well known that, given a crisp partition P of a crisp set X, there exists a unique equivalence relation Q such that P is the quotient set of X by this relation which is defined by:

(32) R T a b , ( ) a′ b′ , ( ) , ( ) T T* a i b j , (
) T* a′ i b′ j , ( ) ,( ) (  )

j 1 = m ∑ i 1 = n ∑ = R T R T R T a b , ( ) a b , ( ) , ( ) φT* a i b j , ( )
j 1 = m ∑ i 1 = n ∑ = R T 1 R T 2 R T a b , ( ) a b , ( ) , ( ) φ a i ( ) φ b j ( ) ⋅ j 1 = m ∑ i 1 = n ∑ φ a i ( ) i 1 = n ∑ φ b j ( ) j 1 = m ∑ ⋅ 1. = = = R T P 1 R T P 2 a
R T P xQy A P x A ∈ y A. ∈ ∧ , ∈ ∃ ⇔ Now, let
and let us assume that the partition P is a finite family of non-empty fuzzy sets, that is and , the fuzzy version of Eq. ( 32) is given by: (33)

Since I is finite, the supremum can be replaced by the maximum which itself can be replaced by a t-conorm S, leading to Q S-T fuzzy relations defined by: (

Given , let us denote the vector in [0, 1] n and , its component defined by:

(35)
Thus, under the condition of proposition 2, we can link the fuzzy relations with the fuzzy equivalence relations , where is Lukasiewicz triangular conorm, i.e. , as follows:

(36)

This approach, which provides a restrictive class of equivalence relations (see [START_REF] Boixader | Fuzzy Equivalence Relations: advanced material[END_REF] for a survey on fuzzy equivalence relations), relies on the same trends as the pioneering works of Bezdek and Harris on likeness relations [3] or Ovchinnikov's on proximity relations [START_REF] Ovchinnikov | An introduction to fuzzy relations[END_REF], which emphasize the definition of fuzzy partitions and study the properties of the associated relations. For example, a fuzzy relation is given in [3] for the Fuzzy C-Means clustering algorithm, where the T L -transitivity is shown from the triangle inequality. Indeed, the reflexivity condition gives the constraint where c is the number of classes and u i (x) the class membership function of the data set X.

As already mentioned, it is also closely related to φ-partitions obtained from an algebraic (or strict) fuzzy partition, as proposed by De Baets and Mesiar [START_REF] Baets | Fuzzy Partitions and their Entropy[END_REF]. Links between a linguistic variable and indistinguishability relations, introduced by Valverde and Trillas [START_REF] Trillas | An inquiry on indistinguishability operators[END_REF], [START_REF] Valverde | On the Structure of F-indistinguishability Operators[END_REF], have been studied by De Soto and Recasens [START_REF] Soto | Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator[END_REF].

Finally, let us also remark that fuzzy relations are a particular case of the parametrized family

I 1, … n , { } = P A i { } i I ∈ = i I A i ∅ ≠ , ∈ ∀ x y , ( ) X X × ∈ Q x y , ( ) , ∀ sup i I ∈ T A i x ( ) A i y ( ) , ( ) . = x y , ( ) X X × ∈ Q S-T x y , ( ) , ∀ S i I ∈ T A i x ( ) A i y ( ) , ( ) . = x X ∈ D x D x i ( ) i I ∈ ∀ x X ∈ ∀ D x i ( ) A i x ( ). = , , Q S L -T R T S L : 0 1 , [ ] 2 0 1 , [ ] → S L u v , ( ) min u v 1 , + ( ) = x y , ( ) X X × ∈ Q S L -T x y , ( ) S L i I ∈ T A i x ( ) A i y ( ) , ( ) = , ∀ S L i I ∈ T D x i ( ) D y i ( ) , ( ) min T D x i ( ) D y i ( ) , ( ) i I ∈ ∑ 1 , ⎝ ⎠ ⎛ ⎞ = = min R T D x D y , ( ) 1 , ( ) R T D x D y , ( ). = = R T M u i x ( ) i 1 = c ∑ 1 = Q S L -T Q S λ Y -T
where is Yager's t-conorm family, that is , obtained when λ=1. They provide an interesting means to define fuzzy equivalence relations when using linguistic hedges as proposed by De Cock and Kerre [START_REF] Cock | A context-based approach to linguistic hedges[END_REF] (see an example in section 5.3 on the fairy-tale problem). Proof. Let us write the condition for P' to be a φ-partition on X:

Fuzzy partitions

(37

)
Since P is φ-partition on X, we have and, therefore, because J is a partition on I, we have:

(38) S λ Y : 0 1 , [ ] 2 0 1 , [ ] → S λ Y u v , ( ) min u λ v λ + ( ) 1/λ 1 , ( ) = X ℜ ⊂ P A i { } i 1, … n , { } ∈ = X ℜ ⊂ D x i ( ) A i x ( ) = D y i ( ) A i y ( ) = i 1, … n , { } ∈ x y , ( ) X 2 ∈ R T D x D y , ( )
R T I 1, … n , { } = P A i { } i I ∈ = X ℜ ⊂ J J k { } k K ∈ = S*: 0 1 , [ ] 2 0 1 , [ ] → S* u v , ( ) φ 1 -S L φ u ( ) φ v ( ) , ( ) ( ) = S L u v , ( ) min u v 1 , + ( ) = P′ B k { } k K ∈ = B k x ( ) S* i J k ∈ A i x ( ) ( ) = X ℜ ⊂ φ B k x ( ) ( ) k K ∈ ∑ φ S* i J k ∈ A i x ( ) ( ) ( ) k K ∈ ∑ S L i J k ∈ φ A i x ( ) ( ) k K ∈ ∑ min φ A i x ( ) ( ) i J k ∈ ∑ 1 , ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ . k K ∈ ∑ = = = φ A i x ( ) ( ) i J k ∈ ∑ 1 ≤ φ B k x ( ) ( ) k K ∈ ∑ φ A i x ( ) ( ) i J k ∈ ∑ k K ∈ ∑ 1. = =
Remark 8: This proposition will be very useful to build fuzzy partition, on cartesian products, for rule-based agregation as described in section 5.5. [START_REF] Dubois | Fuzzy numbers: an overview[END_REF] (see figure 3) is defined by the following membership function:

Definition 5: An (L-R) fuzzy interval

where , and and L, R two non-increasing left-continuous functions from ]0, 1] to [0, 1[, with and for all .

Proposition 12:

Let T be a strict t-norm with a diagonal section φ. Let be a family of (L-R) fuzzy intervals defined on , with for all , and such that:

Then, for all and all , we have Proof. For all , we have . Now, for all and all we have:

(39) ----------------- -----------------------------

A x ( ) L a x - α ----------- ⎝ ⎠ ⎛ ⎞ if x ]a α -a[, , ∈ 1 i f x a b , [ ], ∈ R x b - β ----------- ⎝ ⎠ ⎛ ⎞ if x ]b b β + [, , ∈ 0 o t h e r w i s e , ⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ = a b ℜ ∈ , a b ≤ α β 0 ∞[ , [ ∈ , L x ( ) 0 > R x ( ) 0 > x ]0 1[ , ∈ a-α b+β 1 a b x P A i { } i 1, … n , { } ∈ a i b i α i β i , , , ( ) LR { } i 1, … n , { } ∈ = = a 1 b n , [ ] ℜ ⊂ L u ( ) R u ( ) φ 1 -1 u - ( ) = = u ]0 1] , ∈ b i a i 1 + α i 1 + , - = b i β i + a i 1 + . = i 1, … n 1 - , { } ∈ x ]b i a i 1 + [ , ∈ L a i 1 + x - α i 1 + -
- ⎝ ⎠ ⎛ ⎞ φ 1 -1 φ R x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ - ⎝ ⎠ ⎛ ⎞ . = i 1, … n , { } ∈ α i 1 + β i = i 1, … n 1 - , { } ∈ x ]b i a i 1 + [ , ∈ φ 1 -1 φ R x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ - ⎝ ⎠ ⎛ ⎞ φ 1 -1 φ φ 1 -1 x b i - β i ------------- - ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ - ⎝ ⎠ ⎛ ⎞ φ 1 -x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ φ 1 -1 α i 1 + b i x - + α i 1 + -
- - ⎝ ⎠ ⎛ ⎞ = = = φ 1 -1 a i 1 + x - α i 1 + ------------------- - ⎝ ⎠ ⎛ ⎞ L a i 1 + x - α i 1 + ------------------- ⎝ ⎠ ⎛ ⎞ . = =
Corollary 2: P is a φ-partition on .

Proof. For all

and for all , . Now, for all and all we have:

(40)

Remark 9: This generic method to build the fuzzy partitions associated with relations is a straightforward consequence of the definition of a φ-partition from an algebraic (or strict) fuzzy partition [START_REF] Baets | Fuzzy Partitions and their Entropy[END_REF]. For the same reason, it can be generalized by using a mapping where h: [0, 1] → [0, 1], is a strictly increasing function with h(0) = 0 and h(1) = 1, such that:

(41)

Fuzzy partition on

Proposition 13: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let and . Let and be two -partitions respectively on X 1 and X 2 . Let be the strict t-norm defined by . Then, such that for all , , is a -partition on Proof. According to proposition 10, for all and , and are, at least, -equivalences. Now, from proposition 9 we know that the relation defined by: (  ) (  )

a 1 b n , [ ] ℜ ⊂ i 1, … n , { } ∈ x a i b i , [ ] ∈ φA j x ( )
j 1 = n ∑ 1 = i 1, … n 1 - , { } ∈ x ]b i a i 1 + [ , ∈ φ j x ( )
j 1 = n ∑ φ R x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ φ L a i 1 + x - α i 1 + ------------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ + φ R x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ φ φ 1 -1 φ R x b i - β i ------------- ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ - ⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ ⎞ + 1 . = = = R T φ' 1 -u ( ) φ 1 -h u ( ) ( ) = h u ( ) h 1 u - ( ) + 1 . = X 1 X 2 × ℜ 2 ⊂ I 1, … n , { } = J 1, … m , { } = P 1 A i { } i I ∈ = P 2 B j { } j J ∈ = φ T*: 0 1 , [ ] 2 0 1 , [ ] → T* u v , ( ) φ 1 -φ u ( ) φ v ( ) ⋅ ( ) = P C i j , ( ) { } i j , ( ) I J × ∈ = x 1 x 2 , ( ) X 1 X 2 × ∈ C i j , ( ) x 1 x 2 , ( ) T* A i x 1 ( ) B j x 2 ( ) , ( ) = φ X 1 X 2 × . x 1 y 1 , ( ) X 1 2 ℜ 2 ⊂ ∈ x 2 y 2 , ( ) X 2 2 ℜ 2 ⊂ ∈ R T 1 D x 1 D y 1 , ( ) R T 2 D x 2 D y 2 , ( ) T D
A few years after his seminal paper on fuzzy set theory, Zadeh introduced the concept of fuzzy meaning [START_REF] Zadeh | Quantitative fuzzy semantics[END_REF].

It can be represented by a mapping M: L→F(X), where F(X) is the set of fuzzy subsets of X, given a relation between terms and numbers. The grade of membership to which x belongs to the meaning of the term l will be denoted .

In [START_REF] Zadeh | Quantitative fuzzy semantics[END_REF], Zadeh introduced also the concept of descriptor set that are extensively used in fuzzy sensors [1][21].

It provides a simple means of representing measurement results by a fuzzy subset of linguistic terms. The conversion from numerical to linguistic representation is called fuzzy linguistic (or symbolic) description or, more simply, fuzzy description and is defined by a mapping D: X→F(L), where F(L) is the set of fuzzy subsets of L,

given the same relation between terms and numbers as the one used for the fuzzy meaning. The grade of membership to which l belongs to the description of the number x will be denoted . Fuzzy description is very close to the representation defined in the scale formalism [START_REF] Finkelstein | Representation by symbol systems as an extension of the concept of measurement[END_REF] and allows the introduction of graduality in the conversion of physical states into fuzzy subsets of terms. Then, using the conventional additive notation for discrete fuzzy subsets, we have:

M Small =
D x l ( ) l L ∈ x X ∈ M l x ( ) D x l ( ). = Remark 12:
The fuzzy description of number x is an element of F(L) and, therefore a vector in [0, 1] n if card(L) = n. Thus, the fuzzy description provides a simple means to build fuzzy equivalence relations on set of numbers. The partitions associated with these equivalence relations are given by the fuzzy meaning of the terms. Indeed, given a set of membership functions with , we can always define a bijection such that, for all , . The use of a set linguistic terms l ∈ L and their fuzzy meaning M l is nothing more than a re-labeling of the membership function which will be more convenient to develop applications.

Definition 6: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let L 1 and L 2 be two sets of linguistic terms whose fuzzy meanings are defined respectively on X 1 and X 2 and denoted , for all and . Let be the strict t-norm defined by .

For all , we will say that T* defines the meaning of a new term «l 1 and l 2 » on as follows:

Definition 7: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let L be a set of linguistic terms whose fuzzy meanings are defined on X and denoted , for all . Let be the t-conorm, isomorphic to Lukasiewicz t-conorm, defined by , where . For all , we will say that S* defines the meaning of a new term «l 1 or l 2 » on as follows:

Definition 8: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let L be a set of linguistic terms. Let be a -partition on X, with the fuzzy meanings of the term l, such that, for all all , there exists and Let defined by

. For all , we will say that N* defines the meaning of a new term «not l» on as follows:

P A i { } i I ∈ = I 1, … n , { } = f:I L → i I ∈ M l A f i ( ) = A i M l l 1 L 1 ∈ l 2 L 2 ∈ T*: 0 1 , [ ] 2 0 1 , [ ] → T* u v , ( ) φ 1 -φ u ( ) φ v ( ) ⋅ ( ) = l 1 l 2 , (
)

L 1 L 2 × ∈ L 1 L 2 × x 1 x 2 , ( ) X 1 X 2 × M l 1 and l 2 , ∈ ∀ x 1 x 2 , ( ) T* M l 1 x 1 ( ) M l 2 x 2 ( ) , ( ) . = M l l L ∈ S*: 0 1 , [ ] 2 0 1 , [ ] → S* u v , ( ) φ 1 -S L φ u ( ) φ v ( ) , ( ) ( ) = S L u v , ( ) min u v 1 , + ( ) = l 1 l 2 , ( ) L 2 ∈ L x X M l 1 or l 2 , ∈ ∀ x ( ) S* M l 1 x ( ) M l 2 x ( ) , ( ) . = M l { } l L ∈ φ M l l 1 L ∈ l 2 L ∈ x X φ M l 1 x ( ) ( ) φ M l 2 x ( ) ( ) + 1 . = , ∈ ∀ N*: 0 1 , [ ] 0 1 , [ ] → N* u ( ) φ 1 -1 φ u ( ) - ( ) = l L ∈ L x X M not l , ∈ ∀ x ( ) N* M l x ( ) (
). = Remark 13: The negation N* can be used with -partitions defined by means of (L-R) intervals as shown in section 4.1. Let us note also that the triplet (S*, T*, N*) obtained by means of the bijection φ satisfies the functional equation of Alsina [START_REF] Calvo | The functional equations of Frank and Alsina for uninorms and nullnorms[END_REF], that is S*(T*(u, v), T*(u, N(v)) = u, for all and , with and two -partitions respectively on X 1 and X 2 . The proof is obvious using proposition 14.

Examples

Fuzzy partitions on ℜ

Figure 4 represents three fuzzy partitions and their associated fuzzy equivalence relations generated according to the principle given in section 4.1. They are respectively obtained with the t-norm T M , with h(x) = x and T M with .

Fig. 4 Examples of fuzzy partitions and their associated equivalence relations.

Comparing colors

Let us analyze a more complex example where colored objects have to be compared. It will be assumed that the color information comes from a sensor based on three photo-detectors recreating the effects of the red, green, blue cones of the human eyes. When the sensor information is normalized, the color space is simply defined as the unit cube (R, G, B). In order to allow a simple description of colors, the luminosity will be separated from the

φ u A i = v B j = A i { } i I ∈ B j { } j J ∈ φ T 500 F h x ( ) π 2 ---x ⎝ ⎠ ⎛ ⎞ 2 sin = -2 -1 0 1 2 -2
T M with h x ( ) π 2 ---x ⎝ ⎠ ⎛ ⎞ 2 sin =
chrominance information by a non linear mapping as shown in figure 5. Fig. 5 From the (R, G, B) cube to the Chrominance-Luminance representation Now let us assume that the Delaunay triangulation of the chrominance plane is used to perform a multi-linear interpolation that defines the 2D-fuzzy meanings of the linguistic terms, as shown in figure 6 for the two terms Red and Grey [START_REF] Benoit | A fuzzy colour sensor[END_REF]. The origin of the chrominance plane is labelled with the term Grey and the luminance information should be used to distinguish grey levels from Black to White. Fig. 6 The 2D-Fuzzy meaning of the terms Red and Grey. This representation verifies the property:

, ( 4 9 )
where L= {Red, Yellow, Grey, Magenta, Cyan, Blue, Green}. ------

- luminance chrominance x 1 x 2 1 1 2 --- - 1 2 --- - 0 3 2 ------- - 3 2 ------- r Y' ----- v Y' ----- b Y' ----- = Y' max r v b , , ( ) = 1 M Red x 1 x 2 1 M Grey x 1 x 2 x 1 x 2 , ( ) X ℜ 2 D x 1 x 2 , ( ) l ( ) l L ∈ ∑ , ⊂ ∈ ∀ M l x 1 x 2 , ( ) l L ∈ ∑ 1 = = x 1 o x 2 o , ( ) x 1 o x 2 o , ( )
o O D Color o ( ) , ∈ ∀ D x 1 o x 2 o , ( )

=

The position of each object in the chrominance plane is represented in figure 7.

Fig. 7 Objects in the chrominance plane

The fuzzy description of the pair (x 1 , x 2 ) associated with each object is given in table 6. For example, we have: Thanks to the property given by Eq. ( 49), a T L -equivalence can be obtained from the following fuzzy relation:

D Color(A) = D (0.
(51

)
When there is no ambiguity, it can be expedient to abbreviate Attribute(o) to o, relying on the context for the determination of whether o stands for an object or for its attribute. Thus, for the sake of readibility, the results

given in table 7 are labelled with the objects instead of their color (e.g. A is used in place of Color(A)).

Grey

F 0 0 R T M o 1 o 2 , ( ) O 2 , ∈ ∀ R T M D Color o 1 ( ) D Color o 2 ( ) , ( ) min D Color o 1 ( ) l ( ) D Color o 2 ( ) l ( ) , ( ) l L ∈ ∑ = min D x 1 o 1 x 2 o 1 , ⎝ ⎠ ⎛ ⎞ l ( ) D x 1 o 2 x 2 o 2 , ⎝ ⎠ ⎛ ⎞ l ( ) , ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ l L ∈ ∑ = This
fuzzy relation is a T L -equality, therefore d = t L o is a metric on the set X, where t L is the additive generator of the t-norm T L , that is t L (u) = 1 -u (see [START_REF] Klement | Triangular Norms[END_REF] for example). Thus, it makes it possible to compare objects in terms of distance in the color space:

Comparing beauty

In [START_REF] Cock | On (un)suitable fuzzy relations to model approximate equality[END_REF], De Cock and Kerre have proposed an interesting example where fairy-tale characters, belonging to the set O = {Snowwhite, Witch, Wolf, Dwarf, Prince, Little-Red-Riding-Hood}, have their beauty compared (see also comments on this paper [START_REF] Bodenhofer | A note on approximate equality versus Poincaré paradox[END_REF], [START_REF] Boixader | On the relationship between T-transistivity and approximate equality[END_REF], [START_REF] Cook | Why T-equivalence relations do not resolve the Poincaré paradox, and related issues[END_REF], [START_REF] Klawonn | Should fuzzy equality and similarity satisfy transitivity? Comments on the paper by M[END_REF], [START_REF] Janis | Resemblance in a nearness[END_REF]). Available information is given by the following table :   Rather than interpreting this table as «the fuzzy sets beautiful, average and ugly in O», as suggested in [START_REF] Cock | On (un)suitable fuzzy relations to model approximate equality[END_REF], we will consider it as the fuzzy linguistic descriptions of the characters' beauty (Let us note that in [START_REF] Cock | On (un)suitable fuzzy relations to model approximate equality[END_REF] the set of the fairy-tale characters is denoted X which has another meaning in this paper

R T M R T M R T M R T M o 1 o 2 , ( ) O 2 , ∈ ∀ d Color o 1 ( ) Color o 2 ( ) , ( ) 1-R T M D Color o 1 ( ) D Color o 2 ( ) , ( ) =
we have a horizontal reading of the table instead of a vertical one. For example, let us write the beauty of the Dwarf as Beauty(Dwarf). It is assumed that Beauty(Dwarf) is an unknown piece of information but whose fuzzy description is known and given by: D Beauty(Dwarf) = 0.10/Beautiful + 0.70/Average + 0.20/Ugly.

(

Since the linguistic description of beauty is given by a human being, it will be supposed that he/she uses fuzzy meanings of Beautiful, Average and Ugly resulting from a non-explicit aggregation of several criteria of beauty and it will be assumed that he/she provides coherent information.

As can be observed, the sum of the grades of memberships for each line of table 8 is equal to one. Therefore, a T L -equivalence is obtained from the following fuzzy relation:

.

(

The resulting table for the fairy-tale characters is given in Table 9 which is exactly the same as the one given in [START_REF] Cock | On (un)suitable fuzzy relations to model approximate equality[END_REF].

As for the color example, a distance can be associated with this equivalence relation in order to compare the beauty of the fairy-tale characters:

. (

As mentioned in section 3. ,( )

l L ∈ ∑ = R T M R T M o 1 o 2 , ( ) O 2 , ∈ ∀ d Beauty o 1 ( ) Beauty o 2 ( ) , ( ) 1-R T M D Beauty o 1 ( ) D Beauty o 2 ( ) , ( ) = R T R S λ Y -T S λ Y
, where h is a linguistic hedge [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF].

(56) Indeed, under the conditions for proposition 2, we have:

(57)

The fuzzy relation built using the membership functions modified by powering hedges is the fuzzy relation raised to the power of the hedge as represented in figure 8.

Fig. 8 Links between powering hedges and fuzzy relations.

Table 10 illustrates the equivalence relation generated when using the membership functions obtained with the linguistic hedges very defined as P 2 . It means that the fuzzy descriptions of the characters' beauty are defined on the set L' = {Very_Beautiful, Very_Average, Very_Ugly} as for example:

D Beauty(Dwarf) = 0.01/Very_Beautiful + 0.49/Very_Average + 0.04/Very_Ugly.

Fairy-tale characters with colored clothes

In order to illustrate a fuzzy equivalence relation on a cartesian product, we will compare fairy tale

) P = α M l x ( ) ( ) M l x ( ) ( ) α = λ 0 R S λ Y -T D x D y , ( ) , > ∀ min T M h l ( ) x ( ) ( ) λ M h l ( ) y ( ) ( ) , λ ( ) l L ∈ ∑ ⎝ ⎠ ⎛ ⎞ 1/λ 1 , ⎝ ⎠ ⎛ ⎞ = min T P 1/λ M l x ( ) ( ) λ P 1/λ M l y ( ) ( ) , ( ) λ l L ∈ ∑ ⎝ ⎠ ⎛ ⎞ 1/λ 1 , ⎝ ⎠ ⎛ ⎞ = P 1/λ R T x y , ( ) ( ) = R S λ Y -T R T µ S α Y -T µ Σ-T P α P α M l x ( ) l L ∈ , M h l ( ) x ( ) l L ∈ , R T D x D y , ( ) R S α Y -T D x D y , ( ) P α R T D x D y , ( ) ( ) = R S α Y -T R S 2 Y -T M R T
characters with regard to their beauty and the color of their clothes. The objects whose color were described in section 5.

Comparing comfort

In this section, the aggregation of temperature and humidity to build comfort information [START_REF] Benoit | Fuzzy sensor aggregation: application to comfort measurement[END_REF] is proposed to illustrate fuzzy partitions on a cartesian product X 1 ×X 2 where X 1 and X 2 are respectively associated with the temperature and the humidity. Let L 1 ={Cold, Cool, Mild, Warm, Hot}, L 2 = {Very_Low, Low, Medium, High} and L = {Comfortable, Acceptable, Uncomfortable} be the sets of linguistic terms associated respectively with the temperature, the humidity and the comfort. As explained in section 4.2, the fuzzy meanings are obtained by the union of cartesian products of membership functions X 1 and X 2 . It can be interpreted as a set of linguistic rules represented in figure 9. The black cell corresponds to the term Comfortable, while the grey and white ones are respectively associated with Acceptable and Uncomfortable. The definition of the term Uncomfortable could be obtained similarly. However, it is simpler to use the negation connective and write: Uncomfortable = not(Comfortable or Acceptable).

The fuzzy meanings of the linguistic terms on X 1 , X 2 and X 1 ×X 2 for T = T P are represented in figure 10

Conclusion

In this paper, fuzzy relations have been introduced. It was shown that fuzzy relations are at least T Dequivalences when T is a strict t-norm with a convex additive generator. Finding the greatest T # t-norm such that fuzzy relations are T # -equivalences is still an open question.

Several examples as close as possible to real problems were proposed to illustrate the interest of this work. For example, it justifies the choice of the operators «sum» and «product» in many rule-based applications. Indeed, when the membership functions define a strict partitioning, the sum and the product are respectively the S* and T* operators when T = T M since . Therefore, it makes it possible to have relations on the cartesian product and therefore, to preserve the associated pseudo-metric.

Fig. 1

 1 Fig. 1 Fuzzy partition in the sense of Ruspini.

Example 1 :

 1 Let E = {a, a'} with a = [0.3 0.2 0.5] and a' = [0.1 0.7 0.2]. The mapping , defined from the pair (E, T M ), is a fuzzy reflexive relation on E. Example 2: Let be the t-norm defined by T P (u, v) = u.v. According to equation (10), we get F = {b, b'} with b = [0.5477 0.4472 0.7071] and b' = [0.3162 0.8367 0.4472]. The mapping , defined from the pair (E, T P ), is also a fuzzy reflexive relation on F.

Remark 6 :Proposition 7 :

 67 Under the assumptions of proposition 6, all relations on E are T D -equivalences on E. Let T = T M where T M is the minimum. Let E be a set of vectors in [0, 1] n . Let be the reflexive fuzzy relation on E. Then, is T L -transitive on E where is the Lukasiewicz t-norm, that is .

Example 3 :

 3 Let E 1 = {a, a'} with a = [0.36 0.48 0.80] and a' = [0.48 0.60 0.64] and be the relation on E 1 . Let E 2 = {b, b'} with b = [0.28 0.96] and b' = [0.60 0.80] and be the relation on E 2 . The resulting relations are: ,

  b) (a', b) (a, b') (a', b') (a, b) 1.0000 0.9360 0.9728 0.9105 (a', b) 0.9360 1.0000 0.9105 0.9728 (a, b') 0.9728 0.9105 1.0000 0.9360 (a', b') 0.9105 0.9728 0.9360 1.0000

Fig. 3 A

 3 Fig. 3 A (L-R) fuzzy interval.

Example 4 :

 4 Let us assume that a very simple sensor returns the size of a human being. Let the linguistic set be L = {Small, Medium, Tall} and the measurement set X = {1.4, 1.5, 1.6, 1.7, 1.8} where the sizes are given in meters. A possible relation linking the size attributes to the measurements is given in table .

Now, let us

  assume that O = {A, B, C, D, E, F} is a set of six objects to be analyzed by a fuzzy color sensor. Let be the chrominance information associated with an object o ∈ O, that is Color(o) = . The fuzzy sensor provides the fuzzy description of the chrominance information associated with each object,

( 52 )

 52 For example, the color of B is closer to the color of A than that of C because we have d(Color(B), Color(A)) = 0.6887 and d(Color(B), Color(C)) = 0.7485.

Fig. 9

 9 Fig. 9 Linguistic terms associated with comfort.

  Fig. 11

Table 1 :

 1 Fuzzy reflexive relation on E.

	(11)

Proposition 1: All strict t-norms T satisfy:

Proof. Let us write the t-norm using its multiplicative generator.

Table 2 :

 2 Fuzzy reflexive relation on F.

	b

Proposition 5 :

 5 Let T 1 and T 2 be two strict t-norms whose respective additive generators t 1 and t 2 are strictly con-

	vex and diagonal sections are respectively denoted	and . Let E and F be two sets of vectors in [0, 1] n . Let
	and	be two mappings respectively reflexive on E and F. If the additive generators t 1 and t 2 are such that
	is concave then, for all	,	where	,	and
		for all	.
	Proof. If	is concave, then for all		,
		.

  T D is the smallest tnorm. Although this theorem is general, it also quite weak since many reflexive relations are T D -transitive. Then, a second proposition shows that relations are T L -transitive, where T L is the Lukasiewicz t-norm. Another

proof, based on the equality , was proposed in by Bezdek and Harris [3]. A third proposition concerning the transitivity of relations is given. Finally, simulation results are proposed for the family of Frank t-norms. Proposition 6: Let T be a strict t-norm. Let E be a set of vectors in [0, 1] n . Let be the relation reflexive on E. If the additive generator of the t-norm T is strictly convex, then is a fuzzy relation T D -transitive on E where is the smallest t-norm, that is: T D (x, y) = 0 if x≠1 and y≠1, T D (x, y) = min(x, y) otherwise. Proof. According to proposition 2, is a fuzzy relation. Now, we have to show:

8 :

 8 Let T = T P where T P is the product. Let E be a set of vectors in [0, 1] n . Let

				be the reflexive
	fuzzy relation on E. Then,	is	-transitive on E where	is Yager's t-norm, that is
	Proof. We have to show:
			.	(

  holds true.

	Remark 7: The t-norm	and the family	belong to the family of nilpotent t-norms. A t-norm T # is nilpotent
	if it is continuous and each	is a nilpotent element, i.e. there exists	such that
		. Let us also note that	because the family of Yager t-norms is continuous
	with respect to its parameter and strictly increasing. It is known that any nilpotent t-norm T

# is isomorphic to Lukasiewicz t-norm, i.e. there exists a strictly increasing bijection such that for all

Table 3 :

 3 Fuzzy relations on E 1 , E 2 and E 1 ×E 2 .

Table 4 :

 4 Fuzzy relation between the linguistic set L and the measurement set X.

				Small	Medium	Tall
			1.4	1	0	0
			1.5	0.7	0.4	0
			1.6	0.3	0.8	0
			1.7	0.1	1	0.3
			1.8	0	0.8	0.7
	1/1.4 + 0.7/1.5 + 0.3/1.6 + 0.1/1.7 + 0/1.8 and D 1.7 = 0.1/Small + 1/Medium + 0.3/Tall.	( 4 8 )
	Remark 11: The fuzzy meaning and the fuzzy description are two different projections of the same relation. There-
	fore, for all	and all	, we have

  1, 0.8) = 0.1423/Grey + 0.7423/Red + 0.1154/Magenta.

Table 6 :

 6 Fuzzy descriptions of the colors of the six objects

			Red	Magenta	Blue	Cyan	Green	Yellow
	A	0.1423	0.7423	0.1154	0	0	0	0
	B	0.2536	0.0536	0.6928	0	0	0	0
	C	0.1979	0.3979	0	0	0	0	0.4042
	D	0.0381	0.0381	0	0	0	0	0.9238
	E	0.1916	0	0.1042	0.7042	0	0	0
	F	1	0	0	0	0	0	0

Table 8 :

 8 ). More formally, this means that the beauty of one character in O is a fuzzy subset defined on the set L = {Beautiful, Average, Ugly}. In other words, the fairy-tale characters.

		A	B	C	D	E	F
	A	1.0000	0.3113	0.5402	0.0762	0.2465	0.1423
	B	0.3113	1.0000	0.2515	0.0762	0.2958	0.2536
	C	0.5402	0.2515	1.0000	0.4804	0.1916	0.1979
	D	0.0762	0.0762	0.4804	1.0000	0.0381	0.0381
	E	0.2465	0.2958	0.1916	0.0381	1.0000	0.1916
	F	0.1423	0.2536	0.1979	0.0381	0.1916	1.0000
			Table 7:	fuzzy relation for the color example.

Table 9 :

 9 The fairy-tale fuzzy relation.

								R T M
	∀	(	o 1 o 2 ,	) O ∈	2 ,	R T M D Beauty o 1 ( ) D Beauty o 2 ( ) , (	)	min D Beauty o 1 ( ) l ( ) D Beauty o 2 ( ) l ( )
						Snowwhite	Witch		Wolf	Dwarf	Prince	Red-Riding-
									Hood
	Snowwhite		1.00	0.00		0.00	0.10	0.80	0.50
	Witch			0.00	1.00		0.70	0.50	0.20	0.30
	Wolf			0.00	0.70		1.00	0.20	0.00	0.00
	Dwarf			0.10	0.50		0.20	1.00	0.30	0.60
	Prince			0.80	0.20		0.00	0.30	1.00	0.70
	Red-Riding-		0.50	0.30		0.00	0.60	0.70	1.00
	Hood
						5,	fuzzy relations are a particular case of the parametrized family	where

is Yager's t-conorm family. It provides an interesting feature to deal with linguistic hedges based on the powering of the membership functions

[START_REF] Cock | A context-based approach to linguistic hedges[END_REF]

:

Table 10 :

 10 The fairy-tale fuzzy equivalence relation using the linguistic hedge very.

		Snowwhite	Witch	Wolf	Dwarf	Prince	Red-Riding-
							Hood
	Snowwhite	1.00	0.00	0.00	0.01	0.64	0.25
	Witch	0.00	1.00	0.49	0.25	0.04	0.09
	Wolf	0.00	0.49	1.00	0.04	0.00	0.00
	Dwarf	0.01	0.25	0.04	1.00	0.09	0.36
	Prince	0.64	0.04	0.00	0.09	1.00	0.49
	Red-Riding-	0.25	0.09	0.00	0.36	0.49	1.00
	Hood

Table 11 :

 11 2 are considered as fairy-tale characters' clothes according to table 11.As can be noticed, the Prince's beauty was quite similar to Snowwhite's in table 9 but their respective clothes have different colors leading to a low similarity in the cartesian product. Fairy-tale characters' clothes.

	Clothes	Object
	Snowwhite	D
	Witch	F
	Wolf	C
	Dwarf	B
	Prince	E
	Red-riding-Hood	A

Let O = {Snowwhite, Witch, Wolf, Dwarf, Prince, Little-red-riding-Hood}, L 1 = {Beautiful, Average, Ugly} and L 2 = {Red, Yellow, Grey, Magenta, Cyan, Blue, Green}, the fuzzy relation is given by: (58)

When using the t-norm T = T M , we have and, therefore, T* = T P , which leads to the following table.

Let us note that, for the sake of simplicity, the table is labeled with the character's name instead of the pair (Beauty(o), Color(Clothes(o)) where o would be the character's name.

Table 12 :

 12 The fairy-tale fuzzy equivalence relation on the cartesian product.

Table 13 :

 13 . Now, let S 1 , S 2 , S 3 , S 4 be four comfort situations characterized by the following measurements: Fuzzy meanings on X 1 , X 2 and X 1 ×X 2 for T= T P .The fuzzy meanings on X 1 ×X 2 can be used to compute the fuzzy descriptions of the four situations. It leads to the table 14. Comfort situations.

	Very_Low	Low	Medium	High
	Hot
	Warm
	Mild
	Cool
	Cold

Table 14 :

 14 Fuzzy descriptions of the four comfort situations.

Acknowledgements

The authors would like to thank the referees for their valuable comments and Prof. P. Baras and R. David, LAMA, Université de Savoie, for fruitful discussions.

is a fuzzy equivalence relation on

The reflexivity condition leads to:

(43) Proposition 14: Let T be a strict t-norm with a strictly convex additive generator and a diagonal section φ. Let and . Let and be two φ-partitions respectively on X 1 and X 2 . Then, , we have for all :

(44)

Proof. For the sake of simplicity, let us denote Replacing T* and S* by their respective definitions on the left hand side part leads to:

(45)

Similarly, replacing S* and T* on the right hand side part gives:

(46)

Because P 2 is a φ-partition, we have and, thus:

(47)

Remark 10: It should be noted that equation (45) is not the conventional distributivity between T* and S* which is known to hold true only if T*=T M . Indeed, and are not independant since they belong to the same φ-partition. However, it will be very interesting in applications because it allows to define a «linguistic distributivity» between connectives, which makes it possible to aggregate linguistic terms indifferently on the cartesian product X=X 1 ×X 2 or on the sets X 1 or X 2 , as shown in the next sections. ,( ) (  )

) .

i j , ( )