Reducible means

A n variables mean M is said to be reducible in a certain class of means N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}$$\end{document} when M can be represented as a composition of a finite number M0,…,Mr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{0},\ldots ,M_{r}$$\end{document} of means belonging to N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}$$\end{document}, being less than n the number of variables of every Mi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{i}$$\end{document}. In this paper, a basic classification of reducible means is developed and the notions of S-reducibility, a type of analytically decidible reducibility, and of complete reducibility of a mean are isolated. Several applications of these notions are presented. In particular, a continuous and scale invariant weighting procedure defined on a class M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_{2}$$\end{document} of two variables means is extended without losing its properties to the class of reducible means in M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_{2}$$\end{document}.


Introduction
Given a real interval I and an integer number n ∈ ℕ , a function M ∶ I n → I defined on I is a (n variables) mean when it is internal; i.e., when the twofold inequality is satisfied by M for every x 1 , … , x n ∈ I . x i is said to be an effective variable (or effective argument) of M when there exists a pair , ∈ I such that M| x i = ≠ M| x i = .
The number (M) of effective arguments of M will play a capital role along this paper and, unless otherwise agreed, in the notation M x 1 , … , x n it will assumed that (M) = n . Exceptions which will frequently occur are the ith coordinate (projection) means X i x 1 , … , x n ≡ x i , i = 1, … , n . These are the unique n variables means M with (M) = 1.
(1) min{x 1 , … , x n } ≤ M x 1 , … , x n ≤ max{x 1 , … , x n }, 1 3 A continuous mean with (M) = n defined on I is a mean which is a continuous function on I n . The class  (0) (i) of all continuous means defined on an interval I is closed under composition, so that the function defined by: is a member of  (0) (I) provided that M 0 , … , M r ∈  (0) (i) and M 0 = r . Borrowing a concept from Universal Algebra, it can be said that the set  (0) (i) enhanced with composition has the structure corresponding to a clone. Note that An informal rule can be formulated stating that the difficulty of a problem involving a mean M increases with (M) . The theory of mean inequalities, a cornerstone in the studies on means, constitute a good example of this rule. Situated at the very beginning of this theory, the case n = 2 of the Arithmetic mean-Geometric mean inequality turns out to be equivalent to the nonnegativity of x 1 − x 2 2 , x 1 , x 2 ∈ ℝ , but no one similar reduction is possible when n > 2 . A less simple instance of the rule is the problem of defining a scale invariant weighting on a class  n (I) of n variables means. A function  ∶  n (i) × Δ n−1 →  (i) , where Δ m denotes the standard m-simplex and  (i) ⊇  n (i) is another class of means, is said to be a weighting procedure (defined on  n (i) ) when the following conditions are satisfied (cf. [6] for the case n = 2 ): (W1) (M, (1∕n, … , 1∕n)) = M, (W2)  M, e i = X i , where e i = ( i j ) n j=1 ( i j is the Kronecker delta) and X i x 1 , … , x n ≡ x i is the ith coordinate mean.
A weighting procedure can be understood as a generalization of the process of converting the arithmetic mean A n x 1 , … , x n = x 1 + ⋯ + x n ∕n in the weighted arithmetic (or linear) mean L n,(w 1 ,…,w n ) x 1 , … , x n = w 1 x 1 + ⋯ + w n x n , where w 1 , … , w n ∈ Δ n−1 . If the weighting process is covariant with respect to an arbitrary change of scale, then it is said scale invariant, while it is said continuous when, for every M ∈  n (i) , (M, ⋅) is continuous on Δ n−1 . Now, some schemes of composition like Aczél's or Ryll-Nardzewski's iterations of a two variables mean M 2 which are defined on the dyadic fractions of [0, 1] can be, under mild conditions on M 2 , extended to the whole interval [0, 1] and therefore, continuous and scale invariant weighting procedures defined on  (0)  2 (i) (or even on more general classes of two variables means) can be based on them (cf. [6][7][8]). It must be added that the extension of these algorithms to n = 3 is not immediate (cf. [23]), while general algorithms valid for n > 3 are being currently studied.
In view of the difficulty of a problem generally increases with dimension, it seems natural to express a n variables mean as a composition of means with a less number of variables, and then try to solve the problem for these last ones. Accordingly, along this paper a mean M ∈ (i) with (M) = n is said to be reducible in a class  (i) ⊆ (i) , when M can be expressed as a composition of a finite number of means M 0 , … , M r ∈  (i) satisfying M i < n, i = 0, … , r .

Reducible means
Nevertheless, as reasonable as this program may seem to solve a problem involving means, a full implementation of it will require to decide whether a mean M is reducible or not in a given class  (i) , a certainly non trivial problem.
To insert reducibility of means in a suitable context, let us remind that the problem of expressing a continuous function F ∶ ℝ n → ℝ as a composition of a finite number of continuous functions F i ∶ ℝ n i → ℝ, i = 0, … , r , with n i < n , can be traced back at least to the year 1900, when D. Hilbert presented his collection of 23 problems at the International Conference of Mathematicians in Paris.
In the 13th problem of the collection (cf. [4] or Chap. 11 of [20]) the conjecture was implicit that not all continuous functions of three variables can be expressed as a composition of functions of two variables. In 1957, V. I. Arnold showed the conjecture was not true: all function f ∈  (0) (I 3 ) can be represented in the form: where I = [0, 1] and h ij , ij ∈  (0) (I 2 ) [2,3]. Previously, Kolmogorov had proved that, for n ≥ 3 , every continuous function f ∈  (0) (I n ) can be represented in the form: where h i ∈  (0) (I 3 ) and 1i , 2i ∈  (0) (I n−1 ) ( [17], see also [18]). An iteration of (2) and (3), together with the observation that show that every continuous function f ∈  (0) (I n ) can be represented as a composition of functions of two variables. The reader interested in these developments and its many ramifications is referred to Refs. [10,14,24] and to the references therein. It should be observed that, in view of the functions entering into (2) or (3) are not generally means, the above general results turn out to be barely useful in connection with the problem of reducibility of means.
The purpose of this paper is to show that the strategy of reduction of dimension can be successfully implemented to solve some problems. Even if the general problem of reducibility of a mean M in a class  (i) may be undecidable, a decidable type of reducibility, the S-reducibility, is identified. Strategies of reduction of dimension can be fully implemented for S-reducible means. For instance, if M is S-reducible and M i , i = 0, … , r are its reduced means, then the bijective M-affine functions are easily expressed in terms of the bijective M i -affine functions. The topic of inequalities between mean, another case in which the reduction of dimension may lead to a simplification, it is also considered in this paper. Continuous and scale invariant weighting procedures can be constructed on the class of means which can be expressed as a composition of a finite number of 2 variables means. The paper a i = a 1 + a 2 + ⋯ + a n−1 + a n , contains a detailed presentation of this construction. Other related developments, like the identification of certain classes of reducible means and the presentation of the concept of tree of a formula, will hopefully exhibit some intrinsic interest.
The paper is organized as follows. A notation for simple forms of substitution of variables and the concept of lower means are both introduced in Sect. 2 along with other preliminary materials. In Sect. 3, the problem of reducibility of a mean is generally discussed and some important classes of reducible means are identified. A scheme of classification of reducible means constructed by recurrence on a first step reduction formula ("first layer representation") is presented in Sect. 4. There, the concept of S-reducibility arises as a especially simple case in which the first layer representation is a reduced representation. A reducible mean turns out to be a composition of means all which are the result of a specialization of variables in a S-reducible mean, a result whose proof is given in Sect. 5, after the introduction of the labeled tree of a formula representing a composition of functions. Besides of providing a support for concepts like that of "longest sequence of compositions of functions in a formula", these trees are used to prove some combinatorial relationships involving the numbers of variables and functional symbols in a formula. The idea of "structure" of a formula is also easily materialized in terms of its associated tree. Some applications of reducibility are developed in the last three sections of the paper. The concept of complete reducibility is applied in Sect. 7, where the scale invariant weighting problem is considered. Inequalities between reducible means whose reduced representations share the same structure are addressed in Sect. 6, while the family of M-affine functions of a S-reducible mean is studied in Sect. 8. The final Appendix contains a table of the notations employed in the paper.

Preliminaries
Throughout the paper, the symbol  n (i) will denote a given class of n variables means. The exact extension of the class will depend on the context, but the symbol (i) will stand always for ⋃ +∞ n=2  n (i) , a class containing means of every dimension n > 2 . Different notations for other classes of means will be introduced here an there along the paper. For instance,  (k)  n (i), k = 0, 1, … , will denote the class of n variables  (k) means defined on I.
Means are reflexive functions: the equality M(x, … , x) ≡ x is derived by equating all variables in (1). If the inequalities (1) are strict provided that the variables x i are not all equal, then the mean M is said to be strict. Classical means (arithmetic) A, (geometric) G and (harmonic) H are all strict, but the functions at the leftmost and rightmost members of the inequalities (1), named the extremal means min n and max n are not. The same is true of the coordinate means X i . Now suppose that, once increasingly ordered, the n-tuple x 1 , … , x n ∈ I n takes the form (x i 1 , … , x i n ) ; i.e., x i 1 ≤ ⋯ ≤ x i n . For every k = 1, … , n , the kth order mean X (k) n is then defined by: Clearly min n = X (1) n ≤ ⋯ ≤ X (k) n ⋯ ≤ X (n) n = max n and X (k) n = n for every k = 1, … , n.

Reducible means
Recall that the product order "⪯ " in I n is defined by: It is written (x 1 , … , x n ) ≺ (y 1 , … , y n ) when (x 1 , … , x n ) ⪯ (y 1 , … , y n ) but (x 1 , … , x n ) ≠ (y 1 , … , y n ) . A mean M is said to be isotone when preserves the product order in I n ; i.e., when M( When applied on (continuous) means, a series of operations besides of composition return new (continuous) means. For example, the symmetric group S n acts on a class  n (I) of n variables means by returning, for a given n variables mean M ∈  n (i) and ∈ S n , a new mean M with permuted variables. Namely, if M ∈  n (i) and = ( 1 , … , n ) ∈ S n , then M is said to be symmetric when M = M for every ∈ S n (i.e., when {M} is a set invariant under S n ).
The bold type will be often used to denote the set of indices {1, … , n} . Given a subset J = i 1 , … , i k of , the symbol [J] stands for the increasingly ordered k-tuple (i j 1 , … , i j k ) obtained by ordering the indices in J. The compact notation (x j ) [J] will be used instead of A generalization of the action of S n on  n (I) named specialization of variables is obtained by considering in (4) ∈ (= { | ∶ → } ) instead of ∈ S n . Indeed, when ∈ and k ∈ , the variables whose indices belong to the preimage −1 (k) turn out to be all identified with x k in the equality (4). In this regard, two different notations will be introduced, each one corresponding to a function of simple type. On one hand, for a n variables mean M defined on I and an (increasingly) ordered set of indices [i 1 , … , i k ], i j ∈ , j = 1, … , k , ( 1 ≤ k < n ), let us denote by M [i 1 ,…,i k ] to the (k + 1) variables mean defined on I which is obtained by identifying in M the variables x j with j ≠ i l , l = 1, … , k ; i.e., for every x i 1 , … , x i k , u ∈ I . On the other hand, given a n variables mean M and a subset J ⊆ n , M J will denote the specialization of M obtained by identifying the variables in J; i.e.,

3
Observe that M [i 1 ,…,i k ] = M , where ∈ is given by: and that M J = M for a ∈ defined by: In (6) i 0 can be arbitrarily chosen in ⧵ i 1 , … , i k , while in (7) i 0 must belong to i 1 , … , i k . Both types of the specializations of variables just introduced can be iterated. For instance, if J 1 , … , J r ⊆ n are mutually disjoint subsets of n (i.e., J i ∩ J j = ∅ provided that i ≠ j ), the symbol M J 1 ⋯J r will denote the mean obtained from M by identifying the variables in every J i = J i ; i.e., the mean produced by setting x j = u i for every j ∈ J i , ( i = 1, … , r ). In this way, ;u 1 , … , u r and it will be written The mean (M) f ∈  (0)  n (i) and it is named the (mean) conjugate of M by f. Note that (M, f ) ↦ (M) f is a group action when f ∈ Hom(i) , the group of homeomorphism of the interval I onto itself. The class  n (f (i)) of n variables quasilinear means defined on f(i) is derived by conjugacy of the class  n (ℝ) of linear means: a generic member QL n of  n (f (I)) has the form: where f ∶ I → ℝ is an injective and continuous function and the n-tuple of numbers (w 1 , … , w n ) satisfies w i > 0, i = 1, … , n, ∑ n i=1 w i = 1 (so that (w 1 , … , w n ) ∈ Δ n−1 , the standard (n − 1)-simplex). The function f is called the generator function of QL n , while the numbers w i are said to be its weights. It may be sometimes useful to specify the generator function or the weights of QL n or both ones. For example, L n,(w 1 ,…,w n ) and G n,(w 1 ,…,w n ) will denote respectively the n variables linear and geometric means with weights w 1 , … , w n . G n,(w 1 ,…,w n ) is the instance of QL n with generator f = ln ∶ ℝ + → ℝ. In Ref. [1], the lower mean (untermittel) of a n variables analytic mean M was defined as a solution w of the equation: which turns out, under suitable hypotheses on M, to be a unique (n − 1) variables mean M n−1 . This concept reappears much later in Ref. [13] for the case of means defined on ℝ + . There, the mean M n is said to be "type 2 invariant" with respect to the mean M n−1 provided that: for everyx 1 , … , x n−1 ∈ ℝ + . A presentation of similar concepts for means defined on linear spaces has recently arisen in [15]. In this paper, let us consider the solutions u to (or, in other terms, functions implicitly defined by) equations of the form: where M [i 1 ,…,i k ] is defined by (5). The example furnished by M x 1 , … , x n = max x 1 , … , x n and any given ordered set of indices (11) in this case), shows that a solution to (11) may not be a mean. Now, for a 2 variables mean M, the equation M(x, u) = u has the unique solution u = x provided that M is strict. In this case, the mean (x) = x , a coordinate mean, is not strict. In general, it can be shown the following: Proposition 1 If M is a n variables strict mean, i 1 , … , i k is an ordered set of indices with 1 < k < n , and u = (x i 1 , … , x i k ) is a solution to Eq. (11), then is a k variables strict mean. In the case k = 1 , u = x i = x i , the i-th coordinate mean.
Proof The case k = 1 is trivial, so that let us suppose that 1 < k < n . In this case, the specialization M [i 1 ,…,i k ] of the strict mean M turns out to be a strict mean. Now, suppose that is a solution to Eq. (11).
a contradiction which shows that = ( It is similarly proved that = (x i 1 , … , x i k ) ≥ min j x j with equality only in the case that the variables x i 1 , … , x i k , are identical. Thus, is a k variables strict mean. □ A notation for the set of fixed points of a function will be useful in the sequel. Given a set E, a function ∶ E → E and a subset ∅ ≠ A ⊆ E , the set of fixed points of in A will be denoted by A mean solving Eq. (11)  In connection with the existence of lower means, let us pay attention to the inequalities: where [i 1 , … , i k ] is a given ordered set of indices. It follows from these that provided that or, in other terms, that the interval Suppose that this fixed point was unique whichever be (x i 1 , … , x i k ) ∈ I k . Under this assumption, the function 0 = u 0 (x i 1 , … , x i k ) turns out to be a well defined k variables mean. Let us prove that 0 is continuous on I k . In fact, if x (0) ∈ I k and x (l) +∞ l=1 ⊆ I k is a convergent sequence with x (0) as limit point, then ( 0 (x (l) )) +∞ l=1 turns out to be a bounded sequence contained in I, a fact that quickly follows from the twofold inequality: Let us show that ( 0 (x (l) )) +∞ l=1 really converges to 0 (x (0) ) . Indeed, if u 1 and u 2 were two cluster points of ( 0 (x (l) )) +∞ l=1 , then there would exist two subsequences

3
Reducible means respectively converging to u 1 and u 2 , while the continuity of M enables to write and, similarly, was arbitrarily chosen, the continuity at x (0) follows. The continuity on I k of 0 is a consequence of the arbitrariness of x (0) ∈ I k .
Summarizing the above discussion, it can be stated the following: Proposition 2 Let M be a n variables continuous mean defined on I and, for a given 1 ≤ k < n , consider an ordered set of indices [i 1 ,...,i k ]. Assume that, for every

turns out to be a k variables continuous mean defined on I.
A mean satisfying condition (12) is said to have the FUS property. For continuously differentiable means, the proposition is a consequence of the Implicit Function Theorem ( [19]): the hypothesis of uniqueness of the fixed point of implies the global existence (and uniqueness) of the solution to Eq. (11).
Proof See the previous discussion. □

Example 3 The function
where f i ∶ I → ℝ + , i = 1, … , n , are continuous, turns to be a strict continuous mean defined on I. Note that the weighted arithmetic mean is obtained from (13) by taking positive constants f i = W i ∈ ℝ + , i = 1, … , n ; while the rth weighted counterharmonic mean (cf. [11], pg. 245) is the particular instance of (13) in which The mean FH n has the FUS property. In fact, given an ordered set of indices [i 1 , … , i k ] , it can be written: has a unique solution given by: which is new mean of the form (13).

Reducible means
Indeed

∕2 . Hence
This shows that Proposition 2 is not generally true when the FUS property does not hold: This example also shows that Theorem 11 in Ref. [13] is false (unless a hypothesis implying condition (12) was added to its statement).

Remark 5
Under the conditions of Proposition 2, it is not difficult to see that a mean M has the property FUS provided that, for every x ∈ I k , any of the following conditions is fulfilled: M [1,2] x 1 , whence, in view of the strictness of M, all x i j , j = 1, … , k , must be equal each other, and then u 1 = u 2 . This contradiction proves that the set Fix M [i 1 ,…,i k ] , E is unitary.

Reducibility and irreducibility
Let (i),  (i) be two classes of means defined on an interval I satisfying The representation itself will be said to be a reduced representation of M, while the means M 0 , … , M r ∈  (i) appearing in it will be named reduced means of the representation. The discussion contained in the forthcoming paragraphs attempts to clarify these concepts.
For an injective and continuous function f . In this way, reducibility in a class  (i) turns out to be a notion invariant under conjugacy (provided that  (i) is invariant under conjugacy). When all members with > 2 of a class of means (i) turn out to be reducible in the class  (i) , the class (i) itself is said to be reducible (in  (i) ). The class (I) of quasilinear means on an interval I is a relevant example of a reducible class into itself. To see this, first write a generic linear mean L n ∈ (ℝ) ( n > 2 ) in the form: where k ∈ , k < n , and (17) which shows that every linear mean M ∈ (ℝ) with (M) > 2 is reducible in (ℝ) or, in other terms, that (ℝ) is a reducible class into itself. Due to the invariance under conjugacy, the generic quasilinear mean QL n given by (9) turns out to be reducible in then it is clearly reducible in every class of means  2 (i) satisfying  2 (I) ⊇  1 (i) .
In this way, QL n turns out to be reducible in (f (ℝ)) , the class of all quasilinear means defined in f (ℝ).
It should be noted that the reduction (17) is not unique in the sense that L n can be expressed as the composition of several different linear reduced means. Indeed, nonlinear or even discontinuous means may be reduced terms of certain representations of a linear mean L n as, for instance, in the representation: where N and N are two arbitrarily chosen complementary means defined on ℝ ; i.e., Correspondingly, neither are unique the reductions of the quasilinear mean QL n .
Symmetric polynomial means provides another important class of reducible means. Recall (cf. [11], Chap. V) that the rth symmetric polynomial functions e [r] n x 1 , … , x n is given by: where r ∈ and ∑ ! ∏ r j=1 x i j stands for the sum of all terms of the form ∏ r j=1 x i j with i j ∈ , j = 1, … , r . The rth symmetric polynomial mean [r] n x 1 , … , x n is then defined by: Usually, these means are defined on ℝ + or ℝ + 0 (even if they are naturally defined on the whole ℝ when r ∈ is an odd number). Since n (x) turns out to be reducible when r = 1 or r = n . In the remaining cases, the simple equality (cf. [11], Lemma 2, pg. 324) [1]

enable us to write
Now, the function is the (two variables) weighted power mean with exponent r and weight is the (two variables) weighted geometric mean with weigh 1/r. Replacing these means in the last member of the equalities (21) produces n−1 (x1,…,xn−1) Summarizing the above discussion, it can be stated the following:

Proposition 6
The class (i) of quasilinear means on an interval I is a reducible class into itself .  ℝ + , the class of polynomial symmetric means, is reducible.
in the class  ℝ + ∪  2,ℚ ℝ + , where  2,ℚ ℝ + stands for the class of two variables quasilinear means defined on ℝ + whose weights are all rational numbers.
Proof After the discussion preceding the statement of the proposition, it is deduced that  ℝ + is reducible in the class  (0)  ℝ + of continuous means. In view of (19) and (22), it turns out to be that  ℝ + is reducible in Now consider the class  ℚ (ℝ) consisting of all linear means with rational weights. A linear mean L n ∈ (ℝ) with at least one irrational weight turns out to be reducible in (ℝ) but irreducible in  ℚ (ℝ) . This simple example shows that the concept of reducibility crucially depends on the class  (i) . As affirmed in the Introduction, deciding whether a mean M belonging to a class (i) is reducible or not in another class  (i) constitutes, in general, a highly non trivial problem. To illustrate this fact, let us discuss briefly the case presented by the continuous mean: is the unique solution to the equation: and therefore, M 1 = G . Since (22) [r]

the equality (24) holds if and only if
whence √ x 1 x 2 could be expressed as a rational function of the variables x 1 , x 2 , x 3 , an absurdity. It has been proved that M can not be represented as a composition of two means M 0 , M 1 ∈  (0)  ℝ + in the form given by (24). As a consequence of this fact and the symmetry of M, no one representation of M as a composition of two (two variables) means M 0 , M 1 ∈  (0)  ℝ + is possible. Now suppose that M can be represented as a composition of three (two variables) means, say, in the form: Unlike the preceding case, the reduced means of the representation (25) can not be computed by simple substitutions of the variables. To overcome this difficulty, let assume that M 0 , M 1 , M 2 ∈  (1)  ℝ + (cf. [22], Vol. I, Pt. II, Problem 119 a)) and partially differentiate (25) to obtain Since a representation of M as a composition of two means was shown to be impossible, it can be assumed that M 1x x 1 , x 2 ≠ 0 ≠ M 2y x 2 , x 3 and then, from (26) it is derived where Let us show that the partial derivatives of M can not satisfy a relationship like (27). In fact, taking into account that:

Reducible means
it is seen that (27) is satisfied if and only there exists a pair of functions u, v such that for everyx 1 , x 2 , x 3 ∈ ℝ + . Setting x 1 = x 3 in this equality yields: whence Substituting this expression for v x 2 , x 3 in (28) produces, after reordering terms, the equality: Now, setting x 1 = x 2 in this last equality yields: whence Replacing this expression for u x 3 , x 2 in (29) yields whence In this way, from (30) with x 3 = x 1 and (31) it is obtained (29) Other possibilities of representing the mean M by a composition of k ( ≥ 3 ) two variables means can, in principle, be discarded by deriving from the representation formula a relationship among the partial derivatives of M of a sufficiently high order and then show that this relationship is not really fulfilled by M. However, the complexity of the procedure increases speedily with k and its usefulness is circumscribed to sufficiently regular means. Furthermore, reducibility of M is, at best, established in a narrower class of means.

A classification of reducible means
A classification of reducible means based on a reduction process will be described along the following paragraphs. First of all, note that a generic n variables mean M ∈ (I) reducible in a class  (i) can be written in the form: (32) will be named the first layer representation of the reducible mean M. M i is a coordinate mean when M i = 1 , and it should be observed that the fact that the equality M i = n may hold for any i = 1, … , r , is not in contradiction with the notion of reducibility, but simply implies that the mean M i can be reduced further (and finally expressed as a composition of a finite number of variables means with < n ). Furthermore note, on one hand, that all variables are effective in (32) and, on the other, that the class  (i) contains all means M i . The mean M 0 will be named outer mean while the means M i , i = 1, … , r , will be named inner means of the first layer representation (32).

Reducible means
Depending on the nature of the inner means, let us distinguish three mutually exclusive possibilities as follows: (i) M i = n for any i = 1, … , r; (ii) M i < n for every i = 1, … , r , and there exists a pair of overlapping sets of indices J i = b i , … , e i and J k = b k , … , e k ; i.e., J i ∩ J k ≠ ∅; (iii) M i < n for every i = 1, … , r , and the sets of indices Clearly, the above possibilities are also exhaustive. In the case (i), every inner mean M i with M i = n must be, in its turn, a reducible mean. Suppose, for example, that M 1 = n ; then, the first layer representation of M 1 reads as follows: where 2 ≤ s < n and 1 ≤ b 1i < e 1i ≤ n , 1 ≤ M 1i ≤ n for every i = 1, … , s , and therefore, the three possibilities (i), (ii) and (iii) reappear. Since M is reducible, this process can be continued up to the point in which the inequality (M ij ) < n is satisfied by every inner mean M ij .
In the case (iii), the equality ⋃ r i=1 J i = must hold, so that the family J i ∶ i = 1, … , r constitutes a partition of and therefore, there exists a permutation ∈ S n such that In this case, let us say that M is a simply reducible or S -reducible mean.
The first layer representation of a S-reducible mean M is already a reduced representation of M. This is a characteristic shared with those means falling into case (ii) above. Our first result shows that this case corresponds to reducible means resulting from a specialization of variables in a S-reducible mean.

then M is obtained as a specialization of variables in a S-reducible mean
Proof Assume that M ∈ (i) has (32) as its first layer representation and that a pair (at least) of sets of indices Let us make a substitution of the variables in the expression (32) by applying the following algorithm: for every i = 1, … , n and every j = 1, … , r , replace the variable x i in M j (whenever it appears) by the new variable x (i,j) . Denote the resulting mean by M * . Every new variable in M * appears no more that one time in no more , M 2 (x e 1 +1 , … , x e 2 ), … , M r (x e r−1 +1 , … , x e r )).
than one M j , so that M * is S-reducible. Now, in order to count the number of variables in M * , define ∶ × → {0, 1} as follows: Thus (M * ) = ∑ n,r i,j=1 (i, j) . Since (i, j) ≤ 1, i = 1, … , n , j = 1, … , r , and r ≤ n − 1 , it can be written (M * ) = ∑ n,r i,j=1 (i, j) ≤ nr ≤ n(n − 1) . Now, in view of every variable x i do appear in any M j , it is clear that ∑ r j=1 (i, j) ≥ 1 for any i = 1, … , n . Furthermore, in view of the fact that J i ∩ J k ≠ ∅ for at least a pair i, k of indices, there exists i ∈ such that ∑ r j=1 (i, j) ≥ 2 . In this way, On the other hand, Unlike what occurs with a general reducible mean, an analytical determination of the reduced means is always possible when M is S-reducible in a certain class  (i) . Using a notation introduced in Sect. 2, the first layer representation (32) of M assumes the compact form: where J k ∶ k = 1, … , r is a partition of provided that M is S-reducible. Now, fix k ∈ . If, for every i ∈ ⧵J k , the variable x i in both members of (33) is substituted by  The statement of a theorem of classification of reducible means is postponed until the next section, where the concept of height of a formula is introduced.

The tree of a formula
In the previous sections, the information contained in a formula representing a certain composition of means has been presented in the linear form which is the main characteristic of a list (cf. Chap. 2 of [16]). However, the nonlinear structure of a tree turns out to be more apt for a number of purposes, among them, to introduce some parameters useful in describing the complexity of a representation and then prove some related combinatorial results. In order to define this tree, let be given a finite number of functions F i ∶ I n i → I, i = 0, … , r , with n i ∈ ℕ for every i = 0, … , r . It is assumed that all arguments of every F i are effective, so that F i = n i , i = 0, … , r (being defined and the effectiveness of an argument in the same way as it was made for means in the Introduction). When structured as a list, a composition of the functions F i is expressed by a formula consisting in a finite sequence of variables and the functional symbols F i , i = 0, … , r , separated by parentheses which is written in observance of the standard conventions. F 0 will denote the outermost function of the formula . The set of functional symbols in a formula will be denoted by ( ) , while ( ) will denote the set of variables in . Functional symbols and variables may appear repeatedly in a formula and it will be useful to define a related formula in which repetitions are eliminated. Concretely, a new formula R is derived from by replacing the jth occurrence of the symbol F i by F ij and the jth occurrence of the variable x i by the new variable x ij . For the terminology and basic results on Graph Theory employed in this section, the reader is referred to Refs. [5,9,16].
Let us define a graph T( ) with labeled vertices and arcs, named the tree of the formula , by a pair (V(T( )), Γ) , where V(T( )) is a set and Γ ∶ V(T( )) → V(T( )) is a set valued function, together a rule of labeling, as follows: It is easy to see that T( ) is an acyclic and connected graph; i.e., it is a tree with root vertex root(T( )) labeled F 0 , the outermost function of the formula . The variables x i of are the labels corresponding to the terminal vertices (leaves) of the tree T( ) while the functional symbols of serve to label the branch vertices of T( ) . The arcs of T( ) are labeled with the integers 0, 1, 2, ….
The tree T( ) of a simple formula is illustrated by the labeled tree on the left of Fig. 1. The rightmost tree in the figure is a variation of T( ) in which the labeling of the arcs has been replaced by ordering: the first argument in F i is joined to F i by the leftmost arc, the second one is joined by an arc placed at the right of the first and so on. Both representations make use of the planarity of trees and of their natural imbedding in a plane, but an orientation must be given to the plane in order that the second representation may be implemented. Of course, labeling of arcs is at all necessary when all functions F i are symmetric.
It should be observed that the assignment of the tree T( ) to a formula is univocal and that the formula (T) corresponding to a given labeled tree T can be promptly written. In particular, if T i is the subtree of T( ) rooted at the vertex labeled F i , then T i is a subformula of . A series of integer valued functions related to a tree T = (V, Γ) is now presented. By definition, nl (T) is the the number of leaves of the tree T, while its height h (T) is the length of the longest path joining the root ( root(T) ) with a leaf. For example, a formula with h (T( )) = 1 and nl (T( )) = n simply represents a function depending on no more than n variables. The descent des(v) of a vertex v ∈ V(T) is given by card(Γv) , i.e., by the number of subtrees of v. Note that the relationship holds amongst the standard notion of degree deg (v) of a vertex in a graph and the descent des(v).  Fig. 1 The tree of a simple formula holds for the descent of the vertices v of T( ).

Proof
The tree T( ) of a reduced representation of a reducible mean M ∈  n (I) has the following two particular properties:  Observe that these properties really characterize the trees T which come from a reduced representation formula of a reducible mean M ∈  n (i) . Thus, for example, the tree of the formula in Fig. 1 may actually correspond to the reduced representation of a reducible mean M ∈  4 (i).

Proposition 13
Let be a representation formula of a reducible mean M ∈  n (i) and suppose that r + 1 is the number of reduced means in ; then, Proof From (C1) it is obtained and these inequalities combined with (37) yield Moreover, the inequality follows from (C2), and thus (38) turns out to be a straightforward consequence of (39), (40). □ The height h (T( )) of the reduced representation formula of a reducible mean M ∈  n (i) clearly satisfies h (T( )) ≥ 2 , and the theorem of classification postponed in the preceding section is now stated in the following terms: (38) max {r + 2, n} ≤ var ( ) ≤ (n − 1)(r + 1) − r.
(40) nl (T( )) ≥ n Proof Assume that is the reduced representation formula of a reducible mean M with (M) = n . After Proposition 11(ii), the equality h (T( )) = 2 amounts to the same that the first layer representation of M corresponds to any one of the cases (ii) or (iii) (described in the preceding section). Thus, in view of Proposition 7, one of the alternatives (L1) or (L2) must occur. Now, the case (i) must occur when h (T( )) > 2 and an inductive reasoning on h = h (T( )) enable us to show the assertion that M is, in this case, a composition of reducible means N ∈ (i) with h T N ≤ 2 . First consider the case h = 3 . If h T ′ ≤ 2 for any other reduced representation ′ of M, then the assertion holds trivially and thus, it can be assumed that the first layer representation of M contains a certain number of inner means M i with M i = n . Let M i 1 , … , M i k be these means, so that and define M * 0 to be the mean which is obtained from M after replacing every occurrence of M i j in (41)  The completely reducible means in a class  (I) ; i.e., those reducible means M ∈ (I) whose reduced means are all two variables means belonging to  (i) , deserve a special consideration. An iteration of (17) proves that linear means are completely reducible in (ℝ) and, as a consequence, a quasilinear mean

3
Reducible means M ∈ (i) turn out to be completely reducible in (i) . As an iteration of (22) shows, polynomial symmetric means provide an example of complete reducibility in  (0)  ℝ + . If is the reduced representation formula of a completely reducible mean M, then T( ) turns out to be a binary tree, so that n i = 2 for every i = 0, … , r . As a consequence ∑ r i=0 n i = 2(r + 1) , and Propositions 11(i) and 12 yield: Setting n = 3 in the inequalities (38) of Proposition 13 yields nl (T( )) = r + 2 , which is consistent with the trivial complete reducibility of a reducible three variables mean.
Proof The necessity of the equality (43) was proved in the paragraph preceding the statement of the proposition. In order to prove the sufficiency, assume that (43) holds for the reduced representation formula of M. Then, Propositions 12 and (i) yield and hence In view of (C1), n i ≥ 2 for every i = 0, … , r , and therefore, the last equality implies n i = 2 for every i = 0, … , r ; i.e., M is completely reducible. □ Now consider a S-reducible mean M ∈  n (I) whose reduced representation is given by a formula . Clearly, the number of leaves nl (T( )) of T( ) is exactly n. The converse is also true. In fact, if nl T( ) = n , then the leaves of T( ) are exactly x 1 , … , x n and a similar property is enjoyed by every labeled subtree of T( ) rooted in any child vertex of M 0 : there is no pair of equally labeled leaves. Let T 1 , … , T r denote a complete list of these labeled subtrees and define n i = nl T i , i = 1, … , k . Then the subformula T i must reduce to a single variable when n i = 1 or else represent a certain n i variables mean M i . In any case, M can be written in the form where M i = n i . Since the family constituted by the leaves of T 1 , … , T r is a partition of x 1 , … , x n , this equality shows that M is S-reducible.

Inequalities
The structure of a formula is defined as a modification of T( ) obtained by suppressing the labels corresponding to the branch vertices. Fig. 2 below shows the structure of formula in Fig. 1. Inequalities for reducible means possessing an identical structure can be expectably transferred to inequalities between the corresponding reduced means. Consider the case in which M, N ∈  (0)  n (i) are reducible means whose respective formulae 1 , 2 have identical structure and h T 1 = 2 ( = h T 2 ). Despite of its simplicity, this case turns out to be representative of the various results that can be reached. After Theorem 14, if M i , N i , i = 0, … , r , denote the corresponding reduced means, it can be written: Since the last factor in the last member of these equalities changes of sign when u 1 and u 2 vary on ℝ + , M and N turn out to be non comparable means.
When M, N ∈  n (i) are both completely reducible in a class  (i) and share the same structure, the comparison of M and N reduces to comparisons among two variables means. In this regard, let M 2 , N 2 ∈  2 (i) and M (0) n ∶ n ≥ 3 , N (0) n ∶ n ≥ 3 ⊆  2 (i) be given and consider M n , N n ∈  n (i) defined for every n ≥ 3 by and respectively.

Theorem 20
Assume that the following conditions are satisfied: n or N (0) n is an isotone mean; then the inequalities hold among the means M n , N n .

Proof
The inequality (47) for n = 2 holds by condition (i). Assuming that it holds for a certain k ≥ 2 , by virtue of (ii) and (iii) it can be written, in the case in which M (0) k+1 is isotone: while a similar chain of inequalities holds when N (0) k+1 is isotone. This completes the inductive proof of the proposition. □

Example 21
The conditions of Theorem 20 are satisfied by the means then  will turn out to be a continuous and scale invariant weighting (defined on a suitable subclass of completely reducible means and taking values on another subclass). As a matter of fact, let us prove the following:

Theorem 22
A continuous and scale invariant weighting procedure  2 defined on class  2 (i) of two variables means can be extended to a weighting procedure defined on the class (i) of n variables means which are completely reducible in  2 (I) .
The extension is made through (48) and (50). In this last, Φ ∶ Δ n−1 → [0, 1] r+1 is a suitable continuous function which depends only on the structure of M.
Clearly, the lineal function  depends only on the structure of M. Let us distinguish two cases according to r + 1 = n − 1 or r + 1 > n − 1 . In the first of them,  Δ n−1 is a (n − 1)-simplex whose vertices coincide with n vertices of [0, 1] n−1 , so that the point turns out to be an interior point of

Remark 26
It can be shown that all the (r + 1)-tuples w (i) 0 , … , w (i) r for which the equality M w (i) 0 ,…,w (i) r = X i is true can be obtained from (52) in Lemma 23 provided that the path  is made vary on the paths joining the root vertex M 0 and every leaf labeled with x i . On the other hand, note that the mappings B of Lemma 24 and of Lemma 25 can be easily constructed and thus, Theorem 22 is a result of constructive nature. The fact that B and can be taken not only continuous but also very regular maps and, consequently, that a regular function Φ could be constructed, does not represent a real improvement of Theorem . Indeed, for the continuous and scale invariant weighting procedures for two variables means which are known up to now, the function w ↦ M (w) is not even differentiable.

Example 27
Let M ∈  4 (i) be a completely reducible mean with formula given by The vertices of the cube [0, 1] 5 (whose existence is assured by Lemma 23) which are obtained by completing with 0 's the labels of the corresponding arcs are listed below: In association with these vectors, four linear functions  can be constructed as in the proof of Theorem 22. Their corresponding matrices are given by Clearly, even others matrices could be constructed by choosing a different expansion with 0's or 1's of the vectors of labels. Take, for example, the linear function  whose matrix is the first of the list and consider the 3-simplex Δ v 1 , v 2 , v 3 , v 4 with vertices v 1 = (1, 0, 0, 0, 0), v 2 = (1, 1, 0, 1, 1), v 3 = (0, 0, 0, 0, 0) and v 4 = (1, 0, 1, 0, 0) .
Indeed, the Arithmetic mean-Geometric mean inequality yields: where  w 1 , w 2 , w 3 , w 4 = w 1 + w 2 + w 4 , w 2 , w 4 , w 2 , w 2 and Observe that the first and third matrices in the preceding list both have rank 3 while 4 is the rank of the remaining ones. Presumably, a selection of matrices with lower rank yields more simple functions Φ.