Existence of additive utility on positive semigroups: An elementary proof

Annals of Operations Research 80(1998)269 – 279

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Existence of additive utility on positive semigroups: An elementary proof ★ Juan C. Candeal Facultad de Ciencias Económicas y Empresariales, Departamento de Análisis Económico, Universidad de Zaragoza, c/ Doctor Cerrada 1 – 3, E-50005 Zaragoza, Spain E-mail: candeal\@posta.unizar.es

Esteban Induráin and Esteban Olóriz Departamento de Matemática e Informática, Universidad Pública de Navarra, Campus Arrosadía s.n, E-31006 Pamplona, Spain

This paper is concerned with the representability of totally ordered semigroups as subsets of the additive real line (R, +, ≤). We prove in an elementary way the existence of an additive utility function. Keywords: order, semigroups, Archimedean properties, utility AMS subject classification: Primary: 06F05; Secondary: 20M15, 54F05, 92G05

1.

Introduction

Let (S, ∗) be a semigroup equipped with a total ordering à . A classical problem consists in finding a function u : X → R satisfying x à y ⇔ u(x) ≤ u( y) and u(x ∗ y) = u(x) + u( y) (x, y ∈S). Such a function is usually called an additive utility function. A characterization of the existence of additive utility functions was obtained earlier by Hölder in 1901 (see the original paper, Hölder [11], or Birkhoff [4, p. 300]) for the particular case of translation-invariant totally ordered groups. A strengthening of Hölder’s result for translation-invariant totally ordered semigroups was given by Alimov [3] (see also Fuchs [7, pp. 167ff]). Some other approaches to the latter case are those due to Holman [12] and, more recently, to De Miguel et al. [6]. ★

This research has been partially supported by the Government of Navarre, Spain through the project “Análisis Matemático de la Preferencia” (December 1996).

© J.C. Baltzer AG, Science Publishers

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J.C. Candeal et al. y Existence of additive utility

A common feature of all these papers is that of using the translation-invariance axiom (also called monotonicity property or homogeneity law) so as to define the extensive structure considered in each situation. As pointed out in Krantz et al. [13, p.124] such an axiom (likewise others stated in extensive measurement) could fail in practice. To illustrate, let (x, y) denote a commodity bundle consisting of x pairs of slacks and y shirts. Using the natural addition of vectors, let (x, y) ∗ (z, w) = (x + z, y + w). For this interpretation, however, the translation-invariance axiom is easily seen to be violated: an individual is very likely to prefer (3, 0) over (0, 3) and yet prefer (3, 0) ∗ (0, 3) = (3, 3) over (3,0) ∗ (3,0) = (6,0), contrary to the axiom. In the present paper, unlike aforementioned works, we start with a class of totally ordered positive semigroups (called strongly Archimedean) which may or may not be translation-invariant. We observe that for each semigroup (S, ∗) of this class, it is possible to define a bivariate real-valued map F : S × S → R which accomplishes good properties related to both the order à and the operation ∗. Then, by adding two conditions to F, we obtain a full characterization of the existence of an additive utility function for S. Moreover, this utility function is directly defined from F. It is instructive to compare this approach with others already existing in the literature. Taking into account that we shall work with positive semigroups (also called closed extensive structures), we claim that our setup is more general than the previous ones mentioned above (when addressed to such a particular framework of positive semigroups), since the translation-invariance axiom is not required a priori in our context. Also, the additive utility function we reach in this way is directly defined from the bivariate function F, which has an easy definition. The method used in Krantz et al. [13, section 3.3, pp.77 – 81] is based on embedding the semigroup S into the twofold Cartesian product S × S, which becomes a translation-invariant totally ordered group for both a suitable total order and binary operation. Then Hölder’s theorem is applied to reach the desired conclusion. Actually, this technique is essentially the same as that provided by Fuchs [7, pp. 168 – 169] (according to Fuchs’ proof, S is embedded into its group of quotients). The process followed in De Miguel et al. [6] uses a result by Hion [10], which states that every Archimedean, translation-invariant and positive totally ordered semigroup has an additive pseudo-utility function. Then super-Archimedeaness is applied to prove that such a function is actually injective. A variation of Hion’s theorem was also used in Holman [12] to get a new proof of Alimov’s theorem. Therefore, and as far as we can tell, when referring to semigroups our setting is simpler and in some respects more general than the previous ones. Also, the constructions given in the lemmata previous to the main theorem allow us to gradually obtain pseudo-utility functions whose properties improve in each step. Finally, in the main theorem, an additive utility is reached. This presentation features the novelty of providing such intermediate pseudo-utilities unlike the classical results, that are normally devoted to obtaining, directly, an additive utility. In addition, the translation-invariance will only appear in the last theorem, without being an initial necessity.


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Some other results that concern the same type of representation theorems but involve further generalizations (such as the concatenation operation not being closed) of Alimov’s theorem are often encountered in the literature. However, this different class of generality imposes other restrictive axioms (see, e.g., Krantz et al. [13, theorem 4 on p. 45]). Although our main result could have been stated in the particular frameworks of extensive and conjoint measurement (see [8, 9]), the range of its applications is not restricted to them. So, and as pointed out in Krantz et al. [13, p. 124], additive utility is particularly useful in subjective probability and the measurement of risk. Other applications to expected utility can be seen in [5]. Moreover, additive utility theory is often used in collective decision making to study the utilitaristic social welfare functions (see [14]) as well as to synthesize individual judgements involving the associativity functional equation (see [1, 2]). Therefore, we may conclude that additive utility in the context of semigroups has powerful applications in many branches of the social sciences related to preference modelling. 2.

Preliminaries and notations

Let à be a total order on a set X (i.e., à is a reflexive, antisymmetric, transitive and complete binary relation on X). The notation x ≺ y will stand for ¬ ( y à x). (X, à ) is said to be representable (resp., pseudo-representable) if there exists a real-valued function u : X → R such that x à y if and only if u(x) ≤ u( y) (resp., x à y ⇒ u(x) ≤ u( y)), for every x, y ∈X. The function u is then said to be a utility (resp. pseudo-utility) function for à . A semigroup (S, ∗) is a nonempty set S equipped with an associative binary operation that we shall denote by “∗”. For the sake of brevity, we use juxtaposition to denote the composition of elements, i.e., xy = x ∗ y. Moreover, given n ∈N, x ∈S, we times denote by nx = x ∗ n … ∗ x. A semigroup S with an element e such that xe = x = ex for every x ∈S is said to be a monoid. If each element x in a monoid S has an inverse x , i.e., x( x ) = ( x )x = e, then S is said to be a group. A semigroup S is commutative if xy = yx for every x, y ∈S. A semigroup S equipped with a total ordering à is said to be a totally ordered semigroup. A totally ordered semigroup (S, ∗, à ) is called translation-invariant if x à y ⇔ xz à yz ⇔ zx à zy for every x, y, z ∈S. Note that the translation-invariance property is often referred to as homogeneity or monotonicity (see e.g., Krantz et al. [13, p. 73]; Holman [12]). Given a totally ordered semigroup (S, ∗, à ), an element x ∈S is said to be positive (resp., negative) if x ≺ xx (resp., xx ≺ x). Observe that when (S, ∗, à ) is translation-invariant, an element x ∈S is positive (resp., negative) if and only if y ≺ xy (resp., xy ≺ y) for every y ∈S.

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The subset of all the positive (resp., negative) elements in S constitutes the positive (resp., negative) cone of S, which we shall denote by S + (resp., S – ). When (S, ∗, à ) is translation-invariant, the positive and negative cones are indeed semigroups; that is, for all x, y ∈S + (resp., S – ) xy and yx ∈S + (resp., S – ). A totally ordered semigroup (S, ∗, à ) is said to be: (i) positive (resp., negative) if it contains only positive (resp., negative) elements, (ii) additively representable (resp., additively pseudo-representable) if there exists a utility (resp., pseudo-utility) function u : S → R satisfying u(xy) = u(x) + u( y), for every x, y ∈S). A positive semigroup (S, ∗, à ) is said to be: (i) Archimedean if for every x, y ∈S with x ≺ y, there exists n ∈N such that y ≺ nx, (ii) super-Archimedean if for every x, y ∈S with x ≺ y, there exists n ∈N such that (n + 1)x ≺ ny. A totally ordered semigroup (S, ∗, à ) is said to be Archimedean (resp., superArchimedean) if its positive cone S + is Archimedean (resp., super-Archimedean) and also its negative cone S – is Archimedean (resp., super-Archimedean) with respect to the dual order à d , defined by x à d y ⇔ y à x (x, y ∈S ). Remark 1. Note that for the particular case of translation-invariant totally ordered groups, it is enough to define the above properties of being Archimedean or superArchimedean for the positive cone, due to the existence of inverse elements. This fact fails to be true for semigroups, where, in general, one cone is Archimedean or super-Archimedean independently of what happens in the other cone. As an example, consider ([0, + ∞) × R, ∗, àL ) with “∗” defined as the coordinate-wise sum, and the lexicographic ordering àL given by (a, c) àL (b, d) if a < b or a = b, c ≤ d. In this semigroup, the cone S – is Archimedean but S + is not. For the case of translation-invariant totally ordered groups, the existence of an additive utility function was characterized earlier by Hölder in 1901. This key result can be stated as follows: Theorem 1 (Hölder). A translation-invariant totally ordered group (G, +, à) is additively representable if and only if it is Archimedean. Proof. See Hölder [11], or else Birkhoff [4, p. 300].

u

Remark 2. Hölder’s theorem loses its validity if we replace groups by semigroups, where, in general, the Archimedean property does not even guarantee the existence of utility functions: Consider, for instance, the lexicographic semiplane ((0, +∞) × R, ∗, àL ) with “∗” defined as the coordinate-wise sum and àL being the lexicographic


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ordering. It is a well-known fact (see, e.g., Birkhoff [4, pp. 200 – 201]) that the above structure does not admit a utility function, even non-additive. The above example shows that in order to characterize the additive representability of translation-invariant totally ordered semigroups, some property more restrictive than the Archimedean one is necessary. One such condition was introduced in Alimov [3]. (See also Fuchs [7, pp. 167ff] or Skala [16, pp. 88ff].) Some other equivalent conditions may be seen in Holman [12] or in De Miguel et al. [6]. Matching the approaches of these authors, a key result in this context is stated in the following theorem. Theorem 2. (a)

The following assertions are equivalent for a positive translation-invariant semigroup (S, ∗, à ): (i) (S, ∗, à ) is additively representable, (ii) (S, ∗, à ) is super-Archimedean.

(b) In the case of translation-invariant totally ordered groups, we can add the condition of Archimedeaness to the above equivalences. (c)

A translation-invariant totally ordered semigroup (S, ∗, à ) is additively representable if and only if its positive and negative cones are both additively representable.

Proof. See lemma 5, theorem 2 and theorem 4 in De Miguel et al. [6].

u

Remark 3. A translation-invariant super-Archimedean semigroup is Archimedean. However, in the general case, super-Archimedean semigroups may fail to be Archimedean. Consider the following example: Let S = (0, + ∞) × (0, + ∞) endowed with the operation “*” defined by (a, b) ∗ (c, d) = (a, b + d) for all a, b, c, d ∈(0, + ∞) and with the lexicographic ordering “ àL”. S is not Archimedean because (1, 1) àL (2, 1), while n(1, 1) = (1, n) àL (2, 1), (n ∈N). On the other hand, S is super-Archimedean: Observe that if a < c, then 2(a, b) = (a, 2b) àL (c, d), and if b < d, then there exists n ∈N such that (n + 1)b < nd. Hence, (n + 1) (a, b) = (a, (n + 1)b) àL (a, nd) = n(a, d). Although throughout the paper we shall work with totally ordered semigroups, our results could be adapted to totally preordered semigroups. 3.

Additive utility on positive semigroups

According to the previous results, we observe that in order to find additive utilities on totally ordered semigroups, two conditions are crucial; namely, the translationinvariance and the super-Archimedeaness of the semigroup.

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In this section, we shall start with a class of positive semigroups (called strongly Archimedean) for which it is possible to define, in a very easy way, a bivariate function “F ” which carries good properties related to both the order and the binary operation. Then, by requiring this function to satisfy two additional conditions, we shall be able to prove in a simple way the existence of an additive utility function “u” which closely depends on “F ”. Actually, these conditions characterize the additively representability of the semigroup. It is important to emphasize that, a priori, we do not demand the semigroup to be either translation-invariant or super-Archimedean. So some steps of our construction still remain valid without requiring the semigroup to satisfy such properties. Definition. A positive semigroup (S, ∗, à ) is said to be strongly Archimedean if for every x, y ∈S with x à y, there exists a unique n ∈N such that nx à y ≺ (n + 1)x. Remark 4. (i) A strongly Archimedean positive semigroup is obviously Archimedean. The converse is true if the semigroup is translation-invariant, but it is not true in general. Consider the following example: Endow S = N with the usual sum “+”, and the ordering “ à ” given by 2m(2n + 1) à 2 p(2q + 1) ⇔ m < p or m = p, n ≤ q. It is clear that S is Archimedean. However, it is not strongly Archimedean because (2k + 1)1 à 2 ≺ (2k + 2)1 for any k ∈N. (ii) A strongly Archimedean positive semigroup may fail to be super-Archimedean. Consider the following example: Let S = (0, + ∞) × (0, + ∞) endowed with the operation “∗” given by (a, b) ∗ (c, d) = (a + c, d) for all a, b, c, d ∈(0, + ∞) and with the lexicographic ordering “ àL”. Working with the first coordinate, we realize that S is strongly Archimedean because the positive real line is Archimedean with respect to the natural sum and ordering. On the other hand, S is not super-Archimedean because (1, 1) àL (1, 2) but n(1, 2) = (n, 2) àL (n + 1) (1, 1) = (n + 1, 1). (iii) A super-Archimedean positive semigroup may fail to be strongly Archimedean. An example is the semigroup considered in remark 3. Lemma 1. Let (S, ∗, à ) be a positive semigroup. Then the following assertions are equivalent: (i) S is strongly Archimedean. (ii) There exists a function F : S × S → [0, + ∞) satisfying the following properties: (a) F(x, y) < F(x, z) ⇒ y ≺ z, (x, y, z ∈S). In other words, for every x0 ∈S, the function ux0 : S → R given by ux0(x) = F(x0, x) is a pseudo-utility. (b) F(x, px) = p, (x ∈S, p ∈N). In particular, any strongly Archimedean positive semigroup admits non-trivial pseudo-utility representations.


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Proof. (i) ⇒ (ii) First observe that x ≺ nx for every x ∈X and n ∈N, n > 1. Actually, the existence of k ∈N, k > 1 such that kx à x ≺ 2x would imply that (k + 1)x à x ≺ 2x because (S, ∗, à) is strongly Archimedean and x à x ≺ 2x by positivity of (S, ∗, à). Thus, we would obtain 2k x à x ≺ 2x, which is a contradiction, again because (S, ∗, à) is positive. Now we shall prove that nx ≺ (n + 1)x, for all x ∈S, n ∈N. Suppose that this is not the case. Then there are x ∈S and (the smallest one) n ∈N in such a way that (n + 1)x à nx. Observe that n > 1 because (S, ∗, à) is positive. Once again by positivity, it follows that there exists k ∈{1, 2,…, n – 1} such that (n + k)x à nx ≺ (n + k + 1)x. Observe now that (n + k)x ≺ nx, since (n + k)x = nx would imply that (n + k + 1)x = (n + 1)x à nx, which is a contradiction. Let us consider two cases: (n + k)x à (n – 1)x, or else (n – 1)x ≺ (n + k)x. If the first one holds, then we have (n – 1)x à (n – 1)x ≺ nx and (n + k)x à (n – 1)x ≺ nx ≺ (n + k + 1)x. But this contradicts the fact of S being strongly Archimedean for x, (n – 1)x ∈S, x ≺ (n – 1)x, and the natural numbers n – 1, n + k. The other case also leads to the same contradiction by considering x, (n + k)x ∈S, and n – 1, n + k ∈N. Let us now construct the bivariate map F: By hypothesis, given x, y ∈S with x à y, there exists a unique n ∈N such that nx à y ≺ (n + 1)x. Define F(x, y) = n. In addition, set F(x, y) = 0 whenever y ≺ x. Since nx ≺ (n + 1)x, for all x ∈S, n ∈N, it is plain that F satisfies condition (b). In order to see that F satifies condition (a), let x, y, z∈S, such that F(x, y) < F(x, z). Two cases may occur. If F(x, y) = 0, then y ≺ x. So, since x à z, it holds that y ≺ x à z, and therefore y ≺ z. Suppose now that F(x, y) > 0. Then by definition we have that x à y, x à z. So there exist n, m ∈N, with n < m, such that nx à y ≺ (n + 1)x and mx à z ≺ (m + 1)x. Thus, nx à y ≺ (n + 1)x à mx à z, and therefore y ≺ z. (ii) ⇒ (i) Conversely, if there exists F satisfying (a) and (b), it follows from (b) that px ≺ ( p + 1)x for every x ∈S, n ∈N, so it is enough to prove that S is Archimedean. Thus proceeding, let x ≺ y and let p ∈N be such that F(x, y) < p. It follows from (a) and (b) that y ≺ px. u Lemma 2. A positive semigroup (S, ∗, à) is both strongly Archimedean and superArchimedean if and only if there exists a function F : S × S → [0, + ∞) satisfying the following properties: (a)

F(x, y) < F(x, z) ⇒ y ≺ z, (x, y, z ∈S).

(b) F(x, px) = p, (x ∈S, p ∈N). (c)

For all x, y ∈S with x ≺ y, there exists p ∈N such that F(x, py) > p + 1.

Proof. Define F as in lemma 1. Then F satisfies conditions (a) and (b). Let us show that it also accomplishes (c). Since S is super-Archimedean, given x, y ∈S with x ≺ y, there exists p ∈N such that ( p + 1)x ≺ py. Moreover, since ( p + 1)x ≺ py, there exists q ∈N such that ((q + 1) ( p + 1))x ≺ ( pq)y . Thus, F(x, ( pq)y) ≥ F(x, ( p + 1) (q + 1)x)) = ( p + 1) (q + 1) > pq + 1.


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Conversely, if there exists F satisfying (a), (b) and (c), then S is strongly Archimedean by lemma 1. To see that it is also super-Archimedean, let x, y ∈S be such that x ≺ y, and by (b) and (c), take p ∈N for which F(x , py) > p + 1 = F(x, ( p + 1)x). From (a) it now follows that ( p + 1)x ≺ py. u We now state the main result of our approach. Theorem 3. Let (S, ∗, à) be a positive totally ordered semigroup. The following assertions are equivalent: (i) S is additively representable. (ii) S is strongly Archimedean, super-Archimedean, and translation-invariant. (iii) There exists a function F : S × S → [0, + ∞) satisfying the following properties: (a) (b) (c) (d)

F(x, y) < F(x, z) ⇒ y ≺ z, (x, y, z ∈S). F(x, px) = p, (x ∈S, p ∈N). For all x, y ∈S with x ≺ y, there exists p ∈N such that F(x, py) > p + 1. 0 ≤ F(x, yz) – F(x, y) – F(x, z) ≤ 1 (x, y, z ∈S).

To obtain the proof of theorem 3, we still need another preparatory lemma. Lemma 3. Let (S, ∗, à) be a positive totally ordered semigroup such that there exists a function F : S × S → [0, + ∞) satisfying the conditions (a) to (d) in the statement of theorem 3. Then S has the following properties: (i) For every x, y ∈S, it holds that x ≺ xy, x ≺ yx. (ii) S is commutative. (iii) x ≺ y ⇒ px ≺ py for every p ∈N (x, y ∈S). Proof of lemma 3. (i) We see from (a) to (d) that for all x, y ∈S,

F( x, xy) ≥ F( x, x ) + F( x, y) = 1 + F( x, y) > F( x, y), so y ≺ xy. Similarly, y ≺ yx.

(ii) Let x, y ∈S. Suppose that xy ≠ yx. We can assume, without loss of generality, that xy ≺ yx. Since by lemma 2 S is super-Archimedean, there exists q ∈N such that (q + 1) (xy) ≺ q(yx). But, from (i) it follows that q( yx) ≺ xq( yx) ≺ xq( yx)y = (q + 1) (xy), and we get a contradiction. (iii) Let x, y ∈S such that x ≺ y. Again, since S is super-Archimedean, there exists q ∈N such that (q + 1)x ≺ qy. Now take p ∈N. Three cases may occur. Case 1: If p = q, it follows from conditions (a) and (b) (see proof of lemma 1) that qx ≺ (q + 1)x ≺ qy.


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Case 2: If q < p, from the properties (a) to (d) we obtain

F( x, py) ≥ F( x, ( p − q ) y) + F( x, qy) ≥ ( p − q ) F( x, y) + F( x, ( q + 1) x ) ≥ ( p − q ) F( x, x ) + ( q − 1) = ( p − q ) + ( q + 1) = p + 1 > p = F( x, px ), so F(x, py) > F(x, px), and therefore px ≺ py. Case 3: If p < q, and it would happen that py à px, now taking r ∈N large enough, from case 2 for z = py, t = px it would follow that rz = r( py) à rt = r( px). This contradicts the conclusions of case 2 for x ≺ y. u Proof of theorem 3. The equivalence (i) ⇔ (ii) follows from theorem 2 and remark 4(i). (In fact the condition of being strongly Archimedean is redundant.) (ii) ⇒ (iii). If we define F as in lemma 1, it will already satisfy the conditions (a), (b) and (c), as seen in lemma 2. Let us show that it also satisfies (d): Given x, y, z ∈S, there exist unique p, q ∈N < {0} such that px à y ≺ ( p + 1)x and qx à z ≺ (q + 1)x. (If p (resp. q) equals 0, then we consider the inequality y ≺ x (resp. z ≺ x).) Hence, F(x, y) = p and F(x, z) = q. Since S is translation-invariant, it follows that ( p + q)x à yz ≺ ( p + q + 2)x ⇒ F(x, yz) ∈{ p + q, p + q + 1}. (iii) ⇒ (i) Assume the existence of a map F : S × S → [0, + ∞) satisfying the conditions (a) to (d). Fix an element x0 ∈S and define the function u : S → R as follows: F( x0 , 2 p x ) u( x ) = lim ( x ∈ S ). p → +∞ 2p u is well-defined because the limit exists since the sequence

 F( x0 , 2 p x )    2p   p ∈N is bounded above (F(x0, x) + 1 is an upper bound), and it is non-decreasing because 2F(x0 , 2 p x) ≤ F(x0 , 2 p +1 x), due to property (d) of F. Given x, y ∈S such that x à y, we see from lemma 3(iii) that for each p ∈N, 2 p x à 2 p y and hence F(x0, 2 p x) ≤ F(x0, 2 p y). Thus, u(x) ≤ u( y), and u is a pseudoutility function. Given now x, y ∈S, p ∈N, and because 2 p(xy) = 2 px2 p y by lemma 3(ii), we have 0 ≤ F(x0, 2 p(xy)) – F(x0, 2 p x) – F(x0, 2 p y) ≤ 1. Dividing by 2 p and taking limits, we get u(xy) = u(x) + u( y), so u is additive.


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Let us now prove that u(x) > 0 (x ∈S): If x = x0 , this follows from the fact that F(x0 , 2 px0 ) = 2 p and therefore u(x0 ) = 1. If x0 ≺ x, then by lemma 3(iii), we have that 2 px0 ≺ 2 px, so 1 = u(x0 ) ≤ u(x) and therefore u(x) > 0. Last but not least, if x ≺ x0 , then since S is Archimedean by lemma 1, there exists p ∈N such that x0 ≺ px. By the previous case and taking into account that u is additive, it follows that pu(x) = u( px) > 0 and hence that u(x) > 0. To conclude, it only remains to check the injectivity of u. To do so, note that if x ≺ y then, since S is super-Archimedean by lemma 2, there exists q ∈N such that (q + 1)x ≺ qy. Then (q + 1)u(x) ≤ qu( y), so u(x) ≤ (qy(q +1))u( y) and therefore, since u > 0, u(x) < u( y). u Remark 5. It should be noted that no matter what, three conditions out of (a) to (d) do not imply that the semigroup is translation-invariant. As pointed out above, in our study in the present paper the condition of translation-invariance only appears in the statement of theorem 3, but not as being a prerequisite. Indeed, the proof of the key implication “(iii) ⇒ (i)” does not make use of such a restrictive condition. Remark 6. Hitherto, we have only dealt with positive semigroups. It remains to be seen what happens in the general case. In theorem 2(c) we stated that, under translationinvariance, the additive representability of a totally ordered semigroup is equivalent to the additive representability of its positive and negative cones. Unfortunately, this property is not true in general, that is: The additive representability of the positive and negative cones of a totally ordered semigroup does not imply, in general, the additive representability of the whole semigroup. Let us show an example of the above situation: Consider S = R\{0} with the natural ordering ≤ and the operation “∗” given by

a ∗ b = a + b if a, b < 0 and a ∗ b = | a | + | b | otherwise. The positive cone is S + = (0, + ∞) with the usual sum and order, so it obviously admits additive utilities. The situation for the negative cone S – = (– ∞, 0) is similar. However, the ordering is not translation-invariant on the whole S because (– 5) < ( – 4), but (– 5) ∗ 3 = 8 > (– 4) ∗ 3 = 7. Therefore, (S, ∗, ≤) is not additively representable. Acknowledgements We are grateful to three anonymous referees as well as to the editors, Professor D. Bouyssou and Professor P. Vincke, for their valuable comments and suggestions. References [1] [2]

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