Bernard Guéry

WHAT CONCEPT OF FINALITY TO JUSTIFY MANAGERIAL BEHAVIOR?

Keywords: Purpose, objective, vision, philosophy Keywords: Finality, objective, vision, philosophy

Management tends to present itself as the ultimate end goal of existence, especially in the current context where there has been a hypertrophy of management by objectives within organizations, resulting in circularity and an inversion of ends and means. In this context, this paper aims to explore two distinct conceptions of purpose. The first, an instrumental logic of purpose, allows us to consider objectives as the only end goal of an organization, while the second, transcendent purpose, opens up the possibility of considering the purposes of individuals together with those of the organisation. This second logic of purpose makes it possible to create a conceptual basis for virtue ethics, conceived as a justification of behaviour in the name of an individual agent's purposes. This logic is considered alongside deontology, which precludes purpose as a justification for behaviour, and consequentialism, which focuses solely on the end of an action.

Introduction

Drucker stated in 1954 that "management is not an end in itself" (Drucker, 1993, p. 119). This assertion contended that management could not be a closed system, as it was related to other purposes outside itself. However, management can still be excessive, as Aubert and Gaulejac highlight in the section of their book entitled "managerial system" (2007). Like any ideological system, management is based on a set of principles which are often regarded as absolute, as the '-ism' in 'managerialism' implies. Jorda defines managerialism as follows: "We call managerialism a system of ideas, practices and languages that forms a theory about the world, people and their relationships by applying the principles of administration and management" (Jorda, 2009, p. 150). The '-ism' reflects an absolutization of principles which may no longer fit in with other elements of the worldview, and thus no longer serve to regulate it (ab-solu comes from the Latin for 'without connection'). In a managerial system, management may become disconnected from any other purpose and become its own end, hence the 'violent' character of this mode of framing human activity (Dujarier, 2015, p. 239).

From this perspective, management not only limits alternative ways of understanding interactions, but also comes to limit itself. We can see here that the notion of finality creates a question that deserves to be answered. The aim here is not so much to question the finality of behavior itself, but rather the characteristics of the understandings of finality used to justify behavior, in order to ask ourselves what conception of finality allows us to believe that valid justifications for behavior are not limited to organizational objectives.

Indeed, depending on these differences, the managerial implications will be different. By drawing on philosophy, since the concept of purpose was first developed within this discipline2 , we can interpret the implications for management inherent in different conceptions of purpose. We will therefore explore what is at stake when these distinct conceptions of purpose are used to justify managerial behavior in an organization.

The essence of our work here is to lay bare two different, even opposing, conceptions of purpose. The first, instrumental (or utilitarian) purpose, serves as the philosophical underpinning of the literature on managerial work. We can distinguish a second logic of finality by referring in particular to the thought of Aristotle, Thomas Aquinas and Spaemann. This logic of finality, which, following Spaemann, we will call transcendent finality, firstly allows us to understand how behavior may be justified under the aegis of 'virtue ethics'. Indeed, this transcendent finality constitutes the basis on which virtue ethics is legitimized, by justifying a particular behavior in terms of its conformity with the ultimate end goal of the agent. This can be contrasted with consequentialism, where an action is justified only in terms of its conformity with its own end goal.

After differentiating the concept of purpose from related concepts in management science (objective, challenge), we will define two logics of purpose and three levels of purpose, before applying these to justifications of behavior.

Purpose and its related concepts

In management, purpose often takes the form of an objective or challenge, and it is important to clarify the proximity and distinctions between these two notions.

Purpose and objective

The objective is an omnipresent notion in management. To exemplify this, we need only quote Delavallée: "the notion of objective is at the heart of management" (2005, p. 84); and Morin: "without objectives, there is no management" (1997, p. 157); "objectives and management are inseparable. Managerial thought and management thought [sic] have developed from this concept for centuries" (Morin, 1997, p. 160). Some definitions of management even include the term 'objective' (Chaptal de Chanteloup, 2011, p. 78;Tezenas du Montcel, 1972, p. 199;CENECO, 2000, p. 148). This is why management by objectives "has sometimes become so commonplace that it has become the very definition of management. When we talk about management, which is often contrasted with bureaucracy, we are in fact implicitly talking about management by objectives" (Delavallée, 2005, p. 83).

There is a certain proximity between the notions of objective and purpose. If we look at definitions of the term 'objective', we find the ideas of 'goal' and 'target' on the one hand, and 'will' and 'wish' on the other (Delavallée, 2005, p. 84;Morin, 1997, p. 160;Morsain, 2000, p. 130). This lexical field overlaps with that of the term 'purpose'.

Nevertheless, the two notions are of a different order, and may even seem to be in opposition. To prove this, we must contrast the omnipresence of the notion of the objective against the relatively few references to purpose in the management literature on meaning at work (Berthoin, Antal and Frémeaux, 2013, p. 5). It is as if the managerial objective does not satisfy the need for purpose. In fact, referring to the meaning of 'objective', Delavallée states that "it is not an intention, a purpose or a goal" (2005, p. 84). [START_REF] Falque | Pratiques de la décision: développer ses capacités de discernement [Decision-making practices: developing discernment skills[END_REF] make a distinction between 'objectives', 'goals' and 'purpose'. For them, the objective is "the aim of a precise and positive result" (p. 90), while "the goal is a mental representation of the achievement of a project. Thus, a project contributes to a goal. It is composed of objectives that serve as indicators" (p. 91). We can see here that the goal is more distant, and that the objective is oriented towards it. "We must, however, accept that the goals of an organization are not seen as ends in themselves, but as concrete orientations that contribute to an ideal, a purpose" (p. 92). Thus, objectives are oriented towards and unified by a goal, and goals are oriented towards and unified by a purpose. However, there does not seem to be a difference in nature between these three notions, but rather a difference in the degree of proximity to the action concerned.

Purpose and challenge

At this point, the notion of purpose seems to be similar to that of challenge, if we believe Courbet: "notionally, the difference between challenges and objectives is the following: to overcome the challenges (e.g. the socio-economic survival of the organization or the increase of profits), the decision-makers of the organization set objectives (e.g. to sell more products) which they will try to achieve, for example, through communication devices" [START_REF] Courbet | Media communication: the contributions of social psychology. Towards an epistemological, theoretical and methodological plurality in Media Studies[END_REF]. We therefore have a challenge that could be described as the objective of the objective, except that it is not accompanied by a deadline.

However, this means that the objective and the challenge then remain on the same level, that of concrete situations, as can be seen in the above example: selling products (objective) and increasing profits (challenge) would in fact be two objectives, one short-term, the other long-term, two situations more or less distant from the action and correlated with each other. Purpose, on the other hand, seems to remain at a certain level of abstraction rather than representing a concrete situation; it is thus more akin to 'vision'.

Purpose: a constellation of understandings

Comparing the concept of purpose with that of the objective makes it possible to define the limits of purpose in the negative. However, what is meant by purpose in positive terms remains to be specified. In fact, purpose is not primarily a concept dealt with in HRM. The concept was originally developed in the field of philosophy, by Aristotle, and it is philosophy that has retained a virtual monopoly on debates around its meaning, if we consider reflections on purpose in biological terms as belonging to the philosophy of science [START_REF] Martin-Lagardette | Evolution and finality[END_REF][START_REF] Gallois | Evolution and finality[END_REF]Rouvière, 1951), and the treatment of purpose in Belgian law (Mertens de Wilmars, 2005) as anecdotal. A simple search in the online catalogue of the Bibliothèque nationale de France corroborates this imbalance between references to the topic in philosophy and in management. Indeed, a search using the word 'gestion' (management) or 'management' in the field 'toute la notice' (entire record) and the word 'finalité' (purpose) in the field 'titre' (title) returns only two results, while a similar search using the word 'philosophy' instead of 'management' in the field 'entire record' returns 74 results3 .

Two logics of end

It is therefore necessary to make a detour into philosophy to say something about the irreducible distinction between two logics of end, as this distinction has a significant impact on approaches to management.

The logic of instrumental, or "immanent" (Dewitte, 2010, p. 108), "utilitarian" (p. 105), or "functional" (p. 105) end is characterized as follows: I am fulfilled by this end that is, by means of this finality. This finality is therefore only a means for me, and it is therefore I who am the finality of this finality. Thus, we see that there is a duality of finality: instrumental finality, which is only the end of my action, is distinct from the finality of this finality. By means of an instrumental end that I place before myself, I am my own end. This justifies the term 'immanent finality'. Spaemann quotes Hume to illustrate the immanence of instrumental finality: "We never do one step beyond ourselves" (2015, p. 22).

Transcendent finality, on the other hand, is placed before me. It can be said that it fulfils me. I transcend my "pure factual existence" (Spaemann), in order to "become what I am", as Pindar puts it4 , to reach my "end-form" (cf. Aristotle, 2002, p. 121;1991, p. 167). In this case, there is also a duality of purpose: the end of my action (1) is oriented (or not) to my transcendent purpose as an agent (2).

Therefore, in both cases, we have the agent subject, the purpose (or term) of the action (in some ways comparable to the Stoic skopos), and the end of the agent subject (this end is more like the telos). What differs in both cases is the content of the agent's end. In one case, this is the agent themselves, their 'factual identity' one might say, while in the other it is a form, an essence that the agent has not yet achieved, or has achieved only in potential.

The following short text by Spaemann makes this clear: "Until then, every being had a 'for what' (a Wozu), i.e. aimed at an end that transcended its pure and simple factual existence and in which it found its fulfilment. This end or object was considered a primary reality. The inversion of teleology consists in ignoring such ends, and in trying to found the different domains of being by making them rest on themselves, the intentional and transcendent end being replaced, in this project of self-foundation, by this immanent end: the simple conservation (or maximization of life)" (quoted by Dewitte, 2010, p. 108). One could therefore say that humans are not their own end, since they fulfil themselves (i.e. realize their nature) outside themselves. Thus, my 'end' is identical to my 'form' (Aristotle), I am not (yet) me: my identity is outside my being as it currently exists. This final identity is my plenary being and my actual reason for being. To illustrate this specificity of finality, one could contrast the thoughts of two famous philosophers: "Man [the form-final of man] transcends man [in his (f)actual existence]" (Pascal, 2000, p. 116); and "Become what you are [in potential and calling]" [START_REF] Nietzsche | Ecce Homo: How we become what we are[END_REF]. These two ideas suggest that I am not me, in the sense that my full being, which I am called to join, goes beyond my pure (f)actual existence. A question asked and answered by Aristotle illustrates this idea: "What is the thing that grows? Not what it comes from, but what it is going towards" (2002, p. 121). To give an example of this logic of finality, we could say that a baby is more adult (in potential) than baby, and that it draws its being-baby from the being-adult which 'calls' to it.

It is important to specify that each of the logics of purpose carries a precise anthropological vision. It is clear that in one case I am a perfect and solitary whole, since I am my own end (autarky). In the other, I am a deficient being, I exist in "privation" (Aristotle) of my own being.

It must be added that the substance of this end-form, the promise of plenitude, the content of this being that I lack, integrates the social dimension in a necessary way, so that "the other is the vector of the effectuation of myself" (Ricoeur), and only the social whole to which I belong enjoys autarky. However, it does not follow from this that my transcendent finality is assimilable into the social whole to which I belong. Housset (2008, p. 364) speaks of an "exodus identity", although according to our approach this is less about becoming the other than about becoming other.

To conclude this introduction of the distinctions between the two logics of finality, it should be noted that the logic of instrumental finality will place finality in semantic proximity to the lexical fields of project, interest, and utility, while the logic of transcendent finality is closer to the lexical fields of vocation and destiny. Lalande (1997, p. 353) gives "destiny" or the "destination of a being" as the sixth meaning of the term 'end'. We take up the term 'destiny', with the caveat that if I do not choose my transcendent end (wanting happiness does not depend on me), I at least choose the means to conform to it.

A figure showing the differences between the two logics of purpose illustrates what is at stake in this distinction: The purpose is up to me, I choose it.

I depend on my end, but I choose the means to achieve it.

Proximity to the notions of interest, objective, project.

Proximity to the notions of vocation, 'destiny'.

Assumes I am a perfect, solitary whole.

Assumes I am a deficient being.

Source: Author

Three levels of end

To complete our discussion of different understandings of end before using these as a basis to account for justifications of behavior, we must make an important distinction, following Thomas Aquinas. For Thomas, "there is indeed a double end. The first is the end of the work [finis operis], the other is the end of human life [finis humanae vitae]" (1899, p. 109). Thomas states elsewhere that "it must be considered that sometimes other is the end of the operator [finis operantis], other is the end of the work [finis operis], as the end of the building is the house, while the end of the builder is profit" (1899, p. 129). The end of an action is therefore not necessarily the end that the agent aims to achieve through this action (the end of the construction process is, the completed house, is not the will to live in it or to sell it). For convenience, we will call this the end purpose of the agent, or the intention. It should not be confused with the good, as the natural end of the free agent in general, which Thomas calls the "end of human life" in the text quoted above. We can thus identify three distinct ends, as shown in Figure 2, below:

Source: Author

Spaemann takes up this threefold distinction. He refers to three levels of meaning for the adjective 'good': "1/ The objective, socio-culturally marked goal of action -the finis operis; 2/ The subjective goal of the agentthe finis operantis; and 3/ The objective-subjective goal, the success of life" (1997, p. 8). The distinction between the first two allows us to account for what happens when I voluntarily fail to be efficient, as in the case of pacifist workers who choose to sabotage the weapons they make. "He who voluntarily does something badly indeed achieves precisely his subjective goal by missing his objective goal" (1997, p. 9). However, we need to consider a third level of purpose, again objective from a certain point of view, in order to determine whether a pacifism-inspired desire to sabotage armaments is a good thing. "We have a third level [...] which allows us to judge again the intention of the person who intentionally does a good or bad thing. [...] It is for this reason and in this third sense that the good technician who makes a bomb badly can be better" 1997, p. 10). This objective purpose of the agent is only thinkable within the logic of the transcendent end: "This goal is the goal of man [as man, and not simply as a craftsman mason or manager, otherwise it would be the subjective purpose of the agent]. Unlike particular goals, it is not 'posed', nor 'worked out', but always already found as the 'in view of' which constitutes our existence, the eudaimonia" (1997, p. 10).

Figure 3: A comparative table of the vocabularies of Thomas Aquinas and Spaemann

End of action: house built

End of agent: housing

Natural end of the agent: the good life

Thomas Aquinas

"Finis operis" "Finis operantis" "Finis humanae vitae"

Spaemann "Objective end of the action" "Subjective end of the agent" "Objectivesubjective end of the agent" Source: Author Our aim here is to show that this objective end of the agent plays the role of a principle of synthesis between efficacy (conformity between the means and the objective end of the action, which we will regard as the criterion of consequentialism) and ethicity (conformity between the agent's subjective end and their objective end as an agent, i.e. the capacity of their intention to fulfil the aim of a good life, which we will regard as the criterion of virtue ethics).

Telos and skopos

One might be tempted to compare the distinction made above with the Stoic distinction between two understandings of 'end', telos (τέλος, which can be translated as 'end') and skopos (σκοπός, which can be translated as 'goal' or 'target'). Indeed, there is a degree of similarity, which it is useful to consider, but the analogy has its limits due to the conceptual framework that gives rise to the Stoic distinction.

Indeed, it is in order to circumscribe the contours of what Epictetus' Manual terms "what depends on us" (Arrien, 2000, p. 160) that the Stoics make this distinction, which then appears as "capital" (Goldschmidt, 1989, p. 146). Reaching a target (skopos) does not depend on us, while aiming towards an end (telos) does. For Harl, "at a date which has not been specified, σκοπός was introduced into the terminology of Stoic morality, alongside τέλος, and precisely to differentiate itself from τέλος: σκοπός may be said to differ from τέλος as aiming differs from success" (Harl, 1961, pp. 451-452). This distinction is illustrated in Epictetus through the metaphor of the arrow: "the shooter (with the bow) must do everything to reach the goal (skopos), but it is this act itself which is, if I may say so, the end (telos) which the shooter seeks, and which corresponds to what we call, when it comes to life, the sovereign good; to strike the goal is only a thing that can be wished for, but it is not a thing worth seeking for its own sake" (Arrien, 2000, p. 155). Cicero also uses archery as a metaphor: "the shooter should do everything to reach the goal, and yet it is the act of doing everything that the purpose can be achieved that would be, if I may say so, its ultimate object, corresponding to what we call, when it comes to life, the sovereign good; whereas the act of striking would not be a thing worth seeking for itself" (2000, p. 18).

In certain situations, the skopos could be brought closer to the end of the action, as an external result. This end of the action, the sought-after consequence5 , serves only as an occasion to aim towards another, more important end. But it is in the conception of the telos that the Aristotelian framework, the conceptual cradle of finality, which constitutes the reference we are seeking here, differs from that of the Stoics. Indeed, there is no similarity between the Stoic telos and transcendent finality or the Aristotelian telos. For the Stoics, the telos is the act itself, insofar as it aims at an end, i.e. it is inhabited by the intention of the end, independently of its result (skopos). There is thus a certain immanence of the Stoic telos, which also holds that the attainment of this end depends on us. This is why the distinction between skopos and telos is limited to connecting the end of an action with the transcendent end of the agent, which is more than an act inhabited by an intention.

This distinction is also limited when we try to connect the end of an action (the finished house) with the purpose of the agent involved in the action (their intention, i.e. to dwell in the house). Indeed, highlighting the Stoic distinction between skopos and telos allows us to consider the possibility of attaining the telos despite failing to attain the skopos: "If our activity is animated by the sole motive of acting, according to our choice of life, in accordance with

Reason and Nature, it attains its end (but not always its goal) at every moment" (Arrien, 2000, p. 66). This decorrelation of the two final instances is not conceivable in the Aristotelian tradition. To take Thomas's example, I cannot live in the house or be paid for building it (purpose of the agent) without having completed it (end of action). This allows us to say that Aristotelian virtue ethics does not exclude consequentialism, but incorporates a concern with consequences as a necessary moment of self-realization.

This lack of similarity with the Stoic distinction makes it possible to highlight two aspects of the logic of transcendent finality. The first is that efficacy (achieving the end of an action) is a condition of ethicity (achieving the agent's purpose), which, as we shall see, contrasts with Stoic logic. Good intentions are not enough in the logic of transcendent finality.

The second aspect concerns normative ethics, which we will discuss in the next section. Evoking virtue ethics (i.e. emphasizing a person's moral progression) does not necessarily mean rejecting the consideration of consequences, unlike Stoicism, which, under the pretext of individual progression, places no moral value on attainment of the skopos. According to the logic of transcendent finality, attaining a consequence is a condition (not merely the fact itself) of attaining transcendent finality. On virtue ethics, which incorporates a consideration of consequences, Thomas Aquinas, an exponent of virtue ethics [START_REF] Elders | La vie morale selon saint Thomas d'Aquin: une éthique des vertus [Moral life according to Saint Thomas Aquinas: an ethics of virtue[END_REF], argues, from a nuanced position, that if consequences are premeditated this will have an influence on how good or bad we can consider an act to be. If they are not premeditated but usually [ut in pluribus] follow from the act itself [per se], then they add to the goodness or badness of the act, whereas if they follow the act incidentally [per accidens] and only in few cases [ut in paucioribus] they should not be considered in any moral assessment of the act (1892, p. 161). These are thus two important aspects of the logic of transcendent finality that emerge from a comparison of this logic with the Stoic distinction between telos and skopos.

Vision as a representation of transcendent purpose

Traces of the logic of transcendent purpose can be found in connotations of the managerial use of the term 'vision'. At first glance, it could be said that 'vision' does indeed have a certain semantic proximity to 'purpose': "The strategic vision generally appears as a goal or a mission whose purpose is to provoke a dynamic in the company, which is reminiscent of the company project" (Smida and Condor, 2001, p. 12). The notion of a 'goal' is present in the definition of vision, thus introducing the idea of purpose, and this makes it possible to 'mobilize' and give 'momentum'.

A dynamic present in the transcendent purpose

With regards to the term 'dynamic' (Smida and Condor, 2001, p. 12) and its link with finality, it is interesting to draw a parallel with the teleological inversion theorized by Spaemann, in order to contrast this with the conception of an instrumental finality. Spaemann makes the surprising assertion that in modernity, contrary to our idea of it, "a dynamic ontological structure (in which each being overtakes itself towards the end to which it relates) is replaced by a static structure: everything is now supposed to come down to self-preservation, i.e., to the conservation of what is already there anyway" (Dewitte, 2010, pp. 108-109). For Spaemann, with teleological inversion, "nature becomes the transcendence-free domain of passivity, of the inert affirmation of what already is" (Spaemann, 2015, p. 64). This aligns with the idea that "Aristotelian finality is dynamic" (Lerner, 1969, p. 164). As a result, there is a circularity that loses momentum in the instrumental purpose approach, because instrumental purpose does not have the mobilizing character that transcendent purpose can have (Dewitte, 2010, p. 112). In any case, transcendent purpose manifests itself as a source of dynamics and mobilization.

Vision as a representation of purpose

We must now highlight the link between purpose and vision. It has been said that purpose is the "representation of a desired state" (Morsain, 2000, p. 130). Vision can be defined as the equivalent of this representation in the case of purpose, so that vision is to purpose what representation is to goal. Vision as a representation of finality accommodates particularly well the characteristic of transcendent finality as opposed to the objective: its abstract character, in the sense of being remote from immediate action. However, vision is a term typically used in the vicinity of 'strategy', a term which itself connotes the long term, a certain distance from operational concerns, and which can even have abstract elements in a similar way to finality.

The semantic proximity of vision and transcendent purpose

Moreover, the idea of vision as a representation of transcendent finality is corroborated by the connotations of vision. Indeed, transcendent finality is not subject to choice, and vision is imposed. Unlike sight (in the sense of the action of seeing rather than the physiological faculty), one cannot escape vision. In its original sense of 'the capacity to see', 'vision' is opposed to 'sight'; unlike seeing something, which can be a matter of choice, vision is not optional.

Moreover, the religious connotations of vision link it, within a wide theological tradition, with the qualifier 'beatific' (Thomas Aquinas, 1892, p. 35), referring to the vision of God in the hereafter, or with the qualifier 'mystical', designating a supernatural earthly vision (Grondin, 2008, p. 15). Again, vision in these cases is not a matter of choice, and it should be noted that the act is so closely linked to its object that the same word is used to signify both: 'vision' designates both the act of seeing and the object seen, which, linguistically, suggests the ontological primacy of the object of vision.

A third connotation also situates vision outside the realm of choice: "Originally, vision refers to the representation of a supernatural thing appearing to the sight or mind. [...] This meaning still exists today, but to translate hallucinations or imaginary representations provoked by delirium or dreams" (Smida and Condor, 2001, p. 13). However, by definition, the dream realm is beyond choice. Vision is therefore a term whose connotation invites us to understand denotation as outside the realm of choice: "In the prophetic or religious definition, the degree of control of the representation by the visionary is low. The latter foresees the future without being able to really influence it. The vision is then defined as a deterministic representation of the future" (Smida and Condor, 2001, p. 14). The connection with transcendent purpose through representation is therefore particularly appropriate: the object of the vision, like transcendent purpose, seems to be self-evident.

However, it should be noted that management literature generally suggests an opposing view, "which equates vision more with a representation of a desired future" (Smida and Condor, 2001, p. 14). By modifying the connotations of vision, this literature relegates the term to a longer-term objective, so that there is no difference in nature or plan, only a difference in degree in relation to the objective. Our approach, on the other hand, considers the specificity of vision in relation to a goal. Its purpose would not be a strategic objective; rather, it is characterized by the fact that it is situated on another level, outside the domain of choice, which is not the case with strategic objectives.

Transcendent purpose and management by objectives

The development of the concept of purpose, and in particular the distinction between levels of purpose, shows that the notion of an objective does not exhaust the concept of purpose. Indeed, the natural end of an agent refers to the transcendent end that allows us to go beyond the objective as the sole horizon of a manager's behavior. This finality therefore makes it possible to put the scope of management by objectives into perspective. The idea of the 'natural end of the agent' makes it possible to define the limits of the question 'what am I made for?', based on the idea that happiness is the fundamental ergon of humankind (Aristotle, 1997, p. 57). The idea of the objective does not provide an adequate answer to this question, whereas the natural end of the agent can claim to do so, whatever content we ascribe to this natural end of the agent. A natural end for the agent is only conceivable within the logic of transcendent finality, since according to the logic of instrumental finality the subject is the horizon of its nature, the source of its finality.

What logic of purpose is used to justify behavior?

It is essential to distinguish between the logics of purpose on the one hand and the levels of purpose on the other in order to understand two important ways of justifying managerial behavior. Indeed, each of the two logics of purpose generates a different way of justifying behavior.

The question of how to justify behavior is addressed by normative ethics. Indeed, for Massin, "normative ethics [...] asks what makes an action or a type of action morally right or wrong" (Massin, 2008, p. 3). Normative ethics can be divided into three main streams. Consequentialism and deontology are considered to be "the two great Western moral traditions" (Berger-Douce, 2011, p. 14). The former justifies behavior in terms of consequences, and the latter according to duty (deon). It seems that in the academic literature deontology and consequentialism are presented as opposites, or as complementary. Virtue ethics is proposed as a third way, a possibility of going beyond. Thus, Deslandes writes: "The craze for virtue ethics can be explained by the universality of its notions and their adaptation to the field of business, but also by the failure of other systems, namely deontology, which is linked to Kantian philosophy, and consequentialism, which draws on Mills ' utilitarianism" (2010, p. 103). Therefore, it can be said that "considering the Western philosophical tradition as a whole, virtue ethics is one of the three major currents in ethics, alongside deontology and consequentialism" (Buechler, 2014, p. 2).

The use of these three currents in justifying individual behavior is now well established, particularly in terms of understanding managers' dilemmas in terms of balancing ethics and efficiency. A deontological manager would give primacy to ethics, while a consequentialist manager would give primacy to efficiency. This brief interpretation needs to be clarified. It is here that the distinction between the two logics of purpose makes it possible to account for the tripartition in normative ethics.

Purpose and justification of behavior

We want to show that deontology is a rejection of the use of finality to justify behavior, that consequentialism depends on the logic of instrumental finality, and that virtue ethics is based in the logic of transitive finality.

Deontology would claim to reject the need to consider finality as a relevant moral criterion, as this tradition considers only the data preceding an action (principles, norms, rules, goodwill), and purpose is a prospective motive for action.

In consequentialism, on the other hand, finality is called upon as a relevant criterion for justifying behavior. Finality is perceived as a sought-after consequence, so much so that some theorists speak of teleology when discussing consequentialism and contrasting it with deontology (Cherré, 2011, p. 64;Urasadettan, 2011, pp. 335-336;Courrent, 2003, p. 147). Thus, it can be supposed that consequentialism takes finality into account as part of ethical reasoning, which is not the case with deontology.

However, it must be said that not all teleology is consequentialist. We have shown how transcendent finality differs from a simple consequence. Transcendent finality does not result from action; in fact, the reverse is true: as the name indicates, the consequence 'follows' the action. Therefore, it seems that teleology can take two distinct forms, depending on the logic of the purpose it is based on. Teleology can thus also serve as a foundation for virtue ethics, which situates a 'good' act within the context of the agent's end goal (rather than simply considering the consequence of the action), i.e., the aim of achieving a good life. The end that is considered in consequentialism corresponds to the instrumental end that a subject sets before themselves in the logic of teleological inversion. It is the end of a particular action that is sought, chosen, which acts as a justification for the action posed. In virtue ethics, meanwhile, a non-chosen end, the end of the agent, is made thinkable by developing the logic of the transcendent end. This is not the end of the action, but primarily of the agent, which serves as the justifying criterion for the action.

To further develop the explanation, we can say that the logic of the instrumental end is incompatible with virtue ethics. It does not take into account the idea of becoming better, which is the principle of virtue ethics. Indeed, according to the logic of the instrumental end, the 'goodness' of the acting subject is self-founded insofar as the subject is their own end. They have no reference against which to evaluate a possible distance between their being and their 'being-better', and thus there is no way for them to conceive a 'becoming-better', since to become better is to move towards a form other than that of 'being-same'.

On the other hand, this framework of instrumental finality favors consequentialism. Indeed, we cannot think of finality as anything other than a term of action (a consequence), so it cannot be considered a cause that would have a significant influence on a subject. According to this logic, therefore, there is a decorrelation between acting well and becoming a better person. Consequently, the only ethical aim that can be considered within the framework of the instrumental finality scheme is that of an end of action (a consequence) which is good, that is, a state of affairs outside of myself that can be considered better after my action than before. "Multiplying useful states of the world" (Spaemann, 1997, p. 171) would be the task of consequentialism. It is therefore only the external consequences of my action that can be used to judge its moral goodness, not the fact that the action may have made me a better person.

A concern with becoming better, which is central to virtue ethics, is conceivable according to the logic of the transcendent end. Indeed, the telos as the objective end of the agent constitutes this 'identity of exodus', the 'what you are' of Pindar's formula that it is a question of becoming, and which is the motive, if not the 'motor' of my action (the final cause should not be confused with the efficient cause or the driving cause, but it is the final cause that enables the driving cause as a motor). It is this final form which, once achieved (if it can be achieved), constitutes the 'being-better'.

The following diagram shows the concern of virtue ethics (A). This movement from being to 'being-better' transcends the subject in its pure factual existence, but remains within the subject as considered in its nature. Spaemann speaks of an "objective-subjective" end (1997, p. 8), and it should be added that in virtue ethics as we propose it, movements (B) and (C), i.e. the action and its end (B), realized according to the subjective end of agent (C), are not revoked (which is the case for the Stoic ethics we have discussed). This movement is a necessary condition of movement (A). In terms of the relationship between different ends, we therefore have the construction of the house (B), finalized by the intention to live in it, and the agent's subjective end (C), happiness (A). We have taken the case of the construction of a house, a technical act, in order to clearly differentiate the levels of finality. However, in the case of a purely ethical act, the end of action (B) merges with this 'being-better' (A). It should be noted that an ethical act is not defined as an act without consequences, but as an act whose consequences are not sought for their own sake. For example, an act of the virtue of temperance, such as depriving oneself of food in order to become 'better' (in this case less dependent on one's desires, for example) would lead to a different state of the world: there would be more of this food left for the other members of the group, for example. But what is sought in virtue ethics is not so much this different state of the world (the consequence) as the 'becoming-better', in this case the 'becoming more temperate'.

It would then be necessary to distinguish between the end of the action as the desired result, and the consequences as an accidental and collateral product of this action. This question of purely ethical action is less directly relevant to our topic because we are analyzing virtue ethics within a managerial framework, i.e. integrated into a technical device, rather than in a purely ethical context.

Unlike virtue ethics, consequentialism, based on the logic of the instrumental or 'utilitarian' end (Spaemann), does not conceive of movement (A) as anything other than accidental. Therefore, considered from the perspective of the end of action (B), consequentialist action is transitive, as the building of a house would be (and the world is then considered as an artefact). From another point of view, it is immanent insofar as its end is the intention, the subjective end of the agent, the agent's interest (C). The diffraction of the concept of the end thus allows for the conceptual basis of virtue ethics as a justification for behavior that considers as a main (although not exclusive) criterion the end of the agent himself, beyond the end of the action.

Conclusion

If we again ask the question of what conception of purpose should be used to justify managerial behavior, the answer is that a logic of instrumental purpose limits the horizon of behavioral justifications to intended consequences, which in a managerial context may become confused with the objective. It is in such situations that management can become its own end. On the other hand, a logic of transcendent purpose, beyond providing a conceptual basis for virtue ethics, expands the perspective. Transcendent purpose allows us to envisage a manager's purpose that transcends management, and this transcendent purpose would constitute a reference for action beyond objectives. In relation to this reference point beyond management, objectives would serve as means to an end that would go beyond the objectives themselves. Thus, the transcendent purpose would serve as a regulator, setting limits for management.

In other words, the answer to the question 'what am I made for?' cannot be answered by management. The answer, whatever it may be, lies beyond management. From this perspective, therefore, objectives must be screened for conformity with the transcendent purpose of management, of organizations, and of managers themselves. The transcendent purpose approach can thus help to protect managers from the abuses of management that are highlighted in the criticism of the managerial system mentioned above. The nature of an organization and a manager make it possible to set limits to the absolutism of management requirements. The chosen objective serves a non-chosen purpose; it is therefore a means, and consequently it loses its absolute character. It is in this context that we can conceive of "the company as a reign of ends" (Bowie, 2011, p. 163).

Figure 1 :

 1 Figure 1: The two logics of purpose

Figure 2 :

 2 Figure 2: The three distinct levels of purpose

Figure 4 :

 4 Figure 4: The three levels of finality in consequentialism and virtue ethics

Teacher-researcher at the IPC-Facultés Libres de Philosophie et de Psychologie.

Our aim is not to deal with the question of finality as such, but with the way in which it can be used to justify behaviour. On finality in itself, see Bastit, M. and Wunenburger, J.-J. (Ed.), La Finalité en question, Paris, L'Harmattan, 2000; Duflo, C., La finalité dans la nature, de Descartes à Kant, Paris, Presses Universitaires de France, 1996; Gilson, E., La liberté chez Descartes et la théologie, Paris, Vrin, 2008, see ch.

"Les causes finales et l'idée d'infini"; Johnson, M. R., Aristotle on Teleology, Oxford, OUP, 2006; Laporte, J., "La finalité chez Descartes", Revue d'histoire de la philosophie, 1928, pp. 366-396; Lerner, M.-P., Recherches sur la notion de finalité chez Aristote, Paris, Presses Universitaires de France, 1960; Vaysse, J.-M., Kant et la finalité, Paris, Ellipses, 1999.

Research conducted on 14 June, 2014.

Pythics, II, v. 72. In reality, the translation seems to be "Be as you have come to know yourself"(Paris, Belles Lettres, 1977, pp. 45-46) or "Puisses-tu devenir qui tu es par savoir!" (Paris, LaDifférence, 2004, p. 179). But the popularity of the formula we are borrowing comes from Nietzsche, F., Ecce Homo: comment on devient ce qu'on est,Paris, Denoël, Gonthier, 1971.

The end of an action cannot be equated with its consequence. The end is aimed at within the action, while the consequence is a side effect which is linked to the result but should not be confused with it. This is why we add the epithet 'sought' to the term 'consequence'.


James F Peters Dedicated

W Dyke

I N Herstein

S V Banavar

FREE RINGS AND THEIR GEOMETRIC REALIZATION AS PLANAR CYCLE COMPLEXES. EXTENSION OF J.J. ROTMAN'S FREE GROUP THEOREM

Keywords: 2010 Mathematics Subject Classification. 14F35 (Homotopy theory), 32B25 (Triangulation), 57M05 (Fundamental group, presentations) Free Ring, Free Path Group, Free Path Ring, Free Ring Presentation, Homotopy, Path, Path Cycle, Realization

This paper introduces free rings, free path groups and free path rings. A free ring is a Herstein ring on a free group. A free path group G which is a J.H.C. Whitehead homotopy group free on a collection of continuous maps h : [0, 1] → X (called paths) in a path cycle on a space X. Each path cycle is a sequence of paths with no end path. A free path ring is a Herstein ring on a free path group. The main results in this paper are (1) Every path cycle is realizable as a free path ring (Extension of J.J. Rotman's free group theorem), (2) Every 1-cycle cell complex is realizable as a path cycle and (3) Every 1cycle is realizable as a free ring (this reverbates back to W. Dyke's 1882 work on the derivation of free groups from circuits on simply connected polygons). Inspired by Lupton-Oprea-Scoville digital fundamental groups, a number of conjectures about the geometric realization of free path rings as structures in digital images and as vector fields are given.

From the beginning, geometry has played a key role in the introduction of free groups. A free group G(β, +) is nonempty finite set G of n elements equipped with binary operation (typically written as a +) and a basis β ⊂ G so that every member a ∈ G can be written as a linear combination the generators g ∈ β, i.e., a = g∈β kg, k mod n. Free groups were introduced by W. Dyck in 1882 as a natural biproduct of simply connected polygons P in which every vertex a ∈ P can be reached by traversing the edges between a and a distinguished vertex g ∈ P [START_REF] Dycke | Gruppentheoretischer studien[END_REF]. Later, J.H.C. Whitehead defined free groups in terms of circuits in graphs in which every every vertex can be reached in series of moves from a generator vertex [START_REF] Whitehead | On certain sets of elements in a free group[END_REF] and [START_REF]On equivalent sets of elements in a free group[END_REF]Theorem 4,p. 799] in which circuits on connected surfaces also have corresponding free groups.

This paper also introduces free path rings R(β, +, •), defined in terms of a pair of homomorphisms on a free path group (see Def. 10). Briefly, a free path group is a free group G(β, +) with a basis β ⊂ G that is a collection of paths in a path cycle and there is a homomorphism f : X × β → G. A path cycle is a collection of paths that are continuous maps h : [0, 1] → X on a space X in which the vertexes h(0), h [START_REF] Dycke | Gruppentheoretischer studien[END_REF] are simply connected via h ∈ G and there is no end vertex [START_REF]Path triangulation, cycles and good covers on planar cell complexes. extension of J.H.C. Whitehead's homotopy system geometric realization and E.C. Zeeman's collapsible cone theorems[END_REF], [START_REF] Peters | Descriptive proximal homotopy. Properties and relations[END_REF]App. A]. This leads to an extension of J.J. Rotman's Free Group Theorem. For the extension, see Theorem 4.

Theorem 1. [13, p. 237] If X is a set, then there exists a group F that is free on X. k,ℓ] △ E as a curviliner triangle) on a space K. During its construction over a temporal interval, h(t), k(t ′ ), ℓ(t ′′ ) occur at times t, t ′ , t ′′ , respectively.

h(t) k(t ′ ) ℓ(t ′′ ) K h(0) := ℓ 0 := v 0 , t h = 0, t ℓ = 1 h(1) = k(0) := v 1 , t h = 1, t k = 0 k(1) = ℓ i (0) := v 2 , t k = 1, t ℓ = 0 k h ℓ [h, k, ℓ] △ E Figure 1. [h, k, ℓ] △ E (Curviliner path triangle) → |[h, k, ℓ] △ E| (geometric realization of [h,

Preliminaries

This section briefly visits basic structures underlying free ring presentations of path cycles and path nerve complexes. Recall Definition 1. Path. Let I = [0, 1], the unit interval. A path in a space (S, P ) is a continuous map h : I → S with endpoints h(0) = x 0 , h(1) = x 1 ∈ P and h(t) ∈ S for t ∈ I [14, §2. 1,p.11]. The geometric realization of a path h is denoted |h|.

Remark 1. Paths over Temporal Unit Intervals.

In application, it is often appropriate to consider paths over a temporal unit interval. In that case, the variable t ∈ I would be conveniently thought of as a clock tick and a path h(t) ∈ R for t ∈ [0, 1] would depend on not only the set of points passed through by h but also the time at which a point h(t) is visited [6, §VII.1, p. 174], i.e.,

h[0, 1] → X. t ∈ [0, 1], with h(t) ∈ 0 = start time, t = elapsed time, 1 =end time [0, . . . , t, . . . , 1] ,
and h(t) is an event that occurs at time t. For example, the trajectory of a moving billiard ball E traces a path h(t) with an initial time at t = 0 when the billiard starts moving, visiting a sequence of points (0-cells, or vertexes) as it travels on some surface, and end time t = 1, when E reaches an end point.

Example 1. A sample path triangle (also called a temporal path triangle)

constucted over a temporal interval is shown in Fig. 1. This path triangle models the paths h 1 , h 2 , h 3 traced by a moving object such as a billiard ball during its movements before it reaches a rest state h 3 [START_REF] Dycke | Gruppentheoretischer studien[END_REF]. Temporal path triangles can be found in triangulated video frames [START_REF] Haider | Temporal proximities: Self-similar temporally close shapes[END_REF] in which the changing triangles on triangulated video frame shapes result from the optical flow recorded in the video.

Definition 2. Path Cycle [10, App. A].

A path cycle hCycE in a CW space K is a sequence of continuous maps {h i }

(n-1)[n] i=0 , h i : I → K, [n] = mod n, with h 0 (0) = h n[n] (0) = v 0 ∈ K, h n-1[n] (1) = v n-1[n] ∈ K. Given h i ∈ {h i } (n-1)[n] i=0
, g = h i (0), a basis element in hCycE, so that v h i+k (0) walks forward k vertexes from g to reach h i+k (0)

:= kg = kh i (0) = h i+k (0) ∈ K, k ∈ Z. hi+k (0) hi+k (0) walks back k vertexes from h i+k (0) to reach h i (0) := h (i+k)-k (0) = h i (0). + : K × K → K, is defined by +(v, v ′ ) = move to v ′ via h i (0) = v, h i+k (v ′ ) +(kg, k ′ g) = kg + k ′ g. +(v, 0) = zero moves from v +(kg, 0) = kg + 0 = kg = v. +(v, -v) = move back to v from v +(kg, -kg) = (k -k)g = 0g = g + 0 = v. +(v, -v ′ ) = move back to v ′′ from v ∈ hCycE +(kg, -k ′ g) = (k -k ′ )g = k ′′′ g.

Remark 2. Order of a path cycle.

A k-path cycle has order k and contains k paths.

Example 2.

Path cycle h E in Fig. 2 is a 3-path cycle (its order is 3) and path cycle hCycE ′ in Fig. 5 is a 9-path cycle (its order is 9).

Example 3.

Path cycle h E in Fig. 3 is a 4-path cycle (its order is 3) and path cycle hCycE ′ in Fig. 5 is a 9-path cycle (its order is 9).

Definition 3. 1-Cycle. In a CW space K [17], a 1-cycle E (denoted by cycE) in a CW space K is a collection of path-connected vertexes (0-cells) on edges (1-cells) attached to each other with no end vertex.

Definition 4. Path-Connected. A pair of 0-cells

v, v ′ in a cell complex K is path-connected, provided there is a sequence of paths h 1 , . . . , h k , starting with h 1 (0) := v and ending with h k-1 (1) = h k (0) := v ′ .

Lemma 1. Every path has a geometric realization as an edge.

Proof. From Def. 1, a path h : I → X includes all points h(t), t ∈ [0, 1], i.e., we have the closed set of points

{h(0), . . . , h(t), . . . , h(1)} , for 0 ≤ t ≤ 1,

which is geometrically realized as an edge |h|

= > h(0)h(1) with h(t) ∈ int(|h|), t ∈ (0, 1) (interior of edge |h|).
Theorem 2. Let hCycE be a path cycle in a CW space.

h 1 (1) := h 2 (0) := v 2 h 1 (t) := p h 2 (t) := q h 2 (1) := h 3 (0) := v 3 h 1 (0) := h 3 (1) := v 1 h 3 (t) := r h △ E → v 1 p v 2 q v 3 r K |h △ E|
1 o Every 3-

path cycle has a geometric realization as a 1-cycle triangle. 2 o Every 4-path cycle has a geometric realization as a 1-cycle rectangle. 3 o Every k-path cycle, k ≥ 1 has a geometric realization as a 1-cycle polytope.

Proof.

1 o : Let h E be a 3-path cycle containing paths h, h ′ , h ′′ . From Lemma 1, the geometric realization of a path h is a 1-cell (edge) |h| in a cell complex. Attach the three paths h, h ′ , h ′′ together so that

h ′ (0) := h(1) defines edge > h(1), h ′ (0) h ′′ (0) := h ′ (1) defines edge > h ′ (1), h ′′ (0) h(0) := h ′′ (1) defines edge > h ′′ (1), h(0).
From Def. 4, the vertexes are path-connected in this sequence of three edges (call it E). Hence, E is a 1-cycle that is a triangle, which is the geometric realization of h E (denoted by |h E|).

2 o : Replace h E in 1 o with a 4-path cycle containing paths h, h ′ , h ′′ , h ′′′ and the desired result follows.

3 o : Recall that a planar polytope is a convex hull that is a finite collection of intersecting half spaces [START_REF] Ziegler | Lectures on polytopes[END_REF]. Let hCycE be a k-path cycle. Replace h E in 1 o with a 4-path cycle containing paths h 1 , . . . , h k . From Lemma 1, the geometric realization of a path h is a 1-cell (edge). Assume that the geometric realization each path h ∈ hCycEis a straight edge on a half plane. Since each h i (1) = h i+1 (0) for every h i , h i+1 ∈ hCycE, the pairs of paths with geometrically-realized edges with common vertexes have geometric realizations as interseting half planes. Replace h E in 1 o with the k-path cycle hCycE and the desired result follows.

Example 4. A 3-path cycle h E is shown in Fig. 2. From Theorem 2.1, h E has a geometric realization as a a 1-cycle triangle |h △ E|, also depicted in Fig. 2.

h 1 h 3 h 2 h 4 h 1 (1) := h 2 (0) := v 1 h 2 (1) := h 3 (0) := v 2 h 3 (1) := h 4 (0) := v 3 h 1 (0) := h 4 (1) := v 0 h E → v 0 v 1 v 2 v 3 K |h E|

Free Rings

This section introduces free rings, which are an extension of free groups. Recall (3) There is an element 0 ∈ R so that a + 0 ∈ R for all a ∈ R.

(4) For every a ∈ R, there is an -a ∈ R so that a + (-a) = 0.

(5

) a • b ∈ R. (6) a • (b + c) = a • b + a • c and (7) (b + c) • a = b • a + c • a.
Also recall Definition 6. Homomorphism [3, §7,p. 46]. Let X be a nonempty set and let G, G ′ b groups on X.

A mapping f : G → G ′ is a homomorphism, provided f (ab) = f (a)f (b) for all a, b ∈ G.
In this work, we assume that f :

G → G ′ is
: X × β → G defined by f (a • b) = ka • k ′ b, a, b ∈ X for every subset X in G.

Theorem 3. [8](Path Cycle Free Group Presentation).

Every path cycle in a CW space has a free group presentation.

For the structure of a path cycle, see Def. 2. Let hCycE be a path cycle. Each a ∈ G can be written as kg for g ∈ β, k ∈ Z. Also, let f : G → G, a homomorphism defined by

f (a + b) = f (a) + f (b) = kg + k ′ g = a + b = (k + k ′ )g = k ′′ g, for a, b ∈ G. Hence, from Def.

7, hCycE has a free ring presentation.

A free ring R(β, +, •), introduced in this paper, is the counterpart of a free group G(β, +) with basis beta (see, e.g., [13, p. 237, 1955 edtion]. Unlike a free group, a free ring R is defined in terms of a pair of homomorphisms on a free group.

Definition 8. Free Ring.

Let G(β, +) be a free group with basis β ⊂ G and X be a nonempty set X ⊆ G, generator g ∈ β, a, b ∈ X. Then a ring R(β, +, •) is free on G, provided every map f : X × β → R has a corresponding pair of unique homomorphisms α : X × β → R with respect to + and γ : X × β → R with respect to •, kmod n ∈ Z + , defined by

X ⊆ R, β ⊂ R. α(a, g) = kg ∈ R, γ(a, b) = k ′ g ∈ R, k ∈ Z. α(a + b, g) = α(a, g) + α(b, g) ∈ R for all a, b ∈ X. γ(a, g) = kg, k ∈ Z. γ(a • b) = γ(a, g) • γ(b.g) ∈ R for all a, b ∈ X.
(+, •) is represented in the diagram 1 in Fig. 5. Let α(a, g) = kg ∈ R, γ(a, b) = k ′ g ∈ R, k ∈ Z and let α : X × β → R and γ(a • b) = γ(a, g) • γ(b.g) ∈ R for all a, b ∈ X for X ⊆ G, g ∈ β,
= kg + k ′ g from Def. 8, for all a, b ∈ X, k, k ′ ∈ Z = (k + k ′ )g = k ′′ g ∈ R, k ′′ ∈ Z. γ(a • b) = γ(a, g) • γ(b.g) from Def. 8, = kg • k ′ g from Def. 8, for all a, b ∈ X, k, k ′ ∈ Z = (k • k ′ )g = k ′′′ g ∈ R, k ′′′ ∈ Z.

Free Path Rings

Free path rings are free ring structures built on free path groups.

Definition 9. Free Path Group.

A group G(β, +) with basis set β ⊆ G (a collection of paths in a path cycle hCycE) is a free path group on G, provided there is a homomorphism f :

X × β → G defined by f (h + h ′ ) = kg + k ′ g ∈ G, X ⊂ hCycE ⊂ G ⊆ hCycE, h, h ′ ∈ X for every nonvoid set of paths X ⊆ G.
Free path groups lead to free path rings, each with a structure that is similar to an ordinary free ring.

Definition 10. Free Path Ring.

A ring R(β, +, •) on a free path group G(β, +) with basis set β ⊆ G is a path free ring on G, provided there is a pair of homomorphisms α :

X × β → R defined for + by α(h + h ′ ) = kg + k ′ g ∈ R and γ : X × β → R defined for + by α(h • h ′ ) = kg • k ′ g ∈ R, X ⊂ hCycE ⊂ R, for every pair of paths h ′ ∈ X for X ⊆ G, g ∈ β.
In either case, a reverse move is possible, i.e.,

α(h -h ′ ) = k ′′ reverse moves from h(0) to reach h ′′ (0) kg + (-k ′ )g = (k -k ′ )g = k ′′ g = k ′′ h(0) = h ′′ (0). α(h • h ′ ) = k ′′′ reverse moves from h(0) to reach h ′′′ (0) kg • (-k ′ )g = (k • k ′ )g = k ′′′ g = k ′′′ h(0) = h ′′′ (0).
It is also the case that a free path ring has a zero move (also called the null move), serving as an identity in a free path Ring, which is possible for both + and •.

α(a + b, g) = α(a, g) + α(b, g) ∈ hRngR from Def. 8 = kg + k ′ g from Def. 8, for all a, b ∈ X, k, k ′ ∈ Z = (k + k ′ )g = k ′′ g ∈ R, k ′′ mod n ∈ Z + . γ(a • b, g) = γ(a, g) • γ(b, g)hRngR from Def. 8, = kg • k ′ g from Def. 8, for all a, b ∈ X, k, k ′ ∈ Z = (k • k ′ )g = k ′′′ g ∈ R, k ′′′ mod n ∈ Z + . v 1 v 0 v 5 = v ′ 0 = g ∈ β ⊂ R v ′ j hCycE hCycE ′ v i v 2 v 3 v 4 v ′ 1 Figure 7.
kg + k ′ g = (k + k ′ )g = k ′′ g = k ′′ h(0) = h ′′ (0). •(h, h ′ ) = k ′′′ moves from h(0) to reach h ′′′ (0) kg • k ′ g = (k • k ′ )g = k ′′′ g = k ′′′ h(0) : k ′′ ≤ k ′′ , h ′′ (0) ≤ h ′′′ (0).
In effect, •(h, h ′ ) results in a longer walk that starts at h(0) than the walk +(h, h ′ ) that starts at h(0). This is a distinguishing property of the • ring homomorphism.

Lemma 2. Every path cycle is realizable as a free path group.

Proof. Let hCycE be a path cycle, paths h, h ′ ∈ hCycE, β ⊂ hCycE, g = h(0) ∈ β. From Def. 9, we can define a homomorphism

+(h, h ′ ) = kg + k ′ g = (k + k ′ )g = k ′′ g = k ′′ h(0) ∈ hCycE.
In addition, every move +(h, h ′ ) has a reverse move -kg, i.e.. +(h,

h ′ ) = kg + (-k ′ g) = kg -k ′ g = (k -k ′ )g = k ′′ g = k ′′′ h(0) ∈ hCycE.
With k = 0, we obtain a zero move, which is the identity or null move in hCycE, i.e., +(h, h ′ ) = kg + (0g) = kg + 0 = kg = kh(0) ∈ hCycE. Hence, we obtain G(β, +), a free path group on hCycE.

Lemma 3. Every path cycle is realizable as a free path ring.

Proof. Let hCycE be a path cycle. From Lemma 2, hCycE is realizable as a free path group G(β, +) with basis β and homomorphism +. Similarly, hCycE is realizable as a free path group G(β, +) with basis β and homomorphism •. every h ∈ G has a reverse move -h and there is a zero move +(h, 0) = kg + 0 = kg = h(0) ∈ hCycE. Hence, G is a free path ring.

It is a straightforward step from free path groups to free path rings. Theorem 4. X is a path cycle, then there exists a free path ring R that is free on X.

Proof. Let X be a path cycle. From Lemma 2, X is realizable as a free path group. In addition, from Lemma 3, X is realizable as a free path ring. Replace G(β, +) in Def. 3 with X and the desired result follows.

Theorem 5. Every path cycle is realizable as a 1-cycle.

Proof. From Lemma 1, the geometric realization of a path h is a 1-cell (edge) |h| in a cell complex. Then each of the paths in a path cycle has a geometric realization as an edge. Hence, from Def. 3 (1-cycle), a path cycle has a geometric realization as a 1-cycle.

The converse of Theorem 5 also holds.

Lemma 4. Every 1-cycle cell complex X in a CW space is realizable as a path cycle.

Proof. Let cycE be a 1-cycle on a space X. Let a, b be vertices for an edge > ab in cycE. Then define the path

h : I → X. h(0) = a. h(1) = b.
Hence, every 1-cycle in space X is realizable as a path cycle. Theorem 6. Every 1-cycle cell complex X in a CW space is realizable as a ring that is free on X.

Proof. Let cycE be a 1-cycle on a space X. From Lemma 4, cycE is realizable as a path cycle. Then the desired result is an immediate consequence of Theorem 4. Remark 3. Theorem 4 is a fundamental theorem in homotopy theory, since free path rings have their counterpart in the free groups introduced by W. Dyke cit-eDyck1882freiGruppe. Both free path rings and free groups are a result of geometry. That is, free groups result from cycles in polygons and free path rings result are deriable from 1-cycles in cell complexes.

Conjectures

This section briefly introduces conjectures, which lead to a transition from the results in this paper to closely related results in digital topology from G. Lupton, J. Oprea and N.A. Scoville [START_REF] Lupton | Digital fundamental groups and edge groups of clique complexes[END_REF], [START_REF] Lupton | Subdivision of maps of digital images[END_REF], e.g., Theorem 7. Theorem 5.16 [5, p.22]. Every finitely presented group occurs as the (digital) fundatmental group of some digital image.

Recall that in every triangulated digital image, there is a least one maximal nucleus cluster (MNC) of triangles on a single vertex [7, §1.4, p. 9]. By attaching the edges to the barycenters of the MNC triangles, we obtain a 1-cycle. This leads to Conjecture 1.

Conjecture 1. Every free path ring has a geometric realization as a 1-cycle in a triangulated digital image.

One way to prove Conj. 1 is to define a path over each edge between consecutive barycenters. The end result will be a path cycle, which provides a basis for a free path ring. From Conj. 1, we also have the following three Conjectures.

Figure 2 .

 2 Figure 2. Three Paths that Construct the Sides of a Path Triangle h △ E in a CW space K, leading to its geometric realization |h △ E| as a 2-cell (triangle)

Figure 3 .Example 5 .Figure 4 . k ≥ 1 paths h 1 , h 2 ,Example 6 .

 354126 Figure 3. Four Paths h 1 , h 2 , h 3 , h 4 geometrically realized as the sides of a Path Rectangle h E in a CW space K, leading to its geometric realization |h E| as a 1-cycle (rectangle)

Definition 5 .

 5 Ring [3, §7,p. 46]. A ring is a nonempty set R, a, b, c ∈ R, with operations + and • defined on R such that (1) a + b ∈ R. (2) (a + b) + c = a + (b + c).

Figure 5 .Example 8 .

 58 Figure 5. Free path ring R(β, +, •) on a free group G(β, +) with basis β. In this diagram, black arrows represent inclusion maps, blue arrows are homomorphisms and β = {v 5 = v ′ 0 = g}.

  a, b X are homorphisms for + and •, respectively, defined by α(a + b, g) = α(a, g) + α(b, g) ∈ R from Def. 8

Figure 6 .Example 9 .

 69 Figure 6. Free path ring hRngR(β, +, •) on a free path group hGrpG(β, +) with basis β. In this diagram, black arrows represent inclusion maps, blue arrows are homomorphisms, and hRngR is defined on a free path group hGrpG(β, +) .

Example 10 .

 10 Figure 7. Free path ring R ({hCycE, hCycE ′ : hCycE ∩ hCycE ′ ̸ = ∅} , +, •) on a pair of intersecting path cycles hCycE, hCycE ′ .

Conjecture 2 .Conjecture 3 .Conjecture 4 .

 234 Every free path ring has a geometric realization in a collection of intersecting 1-cyles in a triangulated digital image. Every free path ring has a geometric realization in some triangulated digital image. Every free path ring has a realization as a vector field.

This diagram is a modified version of the gluing diagram introduced by Tane Vergili in [12, p. 3].

The research has been supported by the Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036 and Scientific and Technological Research Council of Turkey (TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211301223.