An Extension Theorem for Conditionally Additive Functions and Its Application for the Equality Problem of Quasi-Arithmetic Expressions

The focus of this paper is the equality problem of quasi-arithmetic expressions. This class is a far generalization of the well-known class of quasi-arithmetic means. One of the main tools in the proof is an extension theorem of real homomorphisms from a subset with a very weak structure (dyadically closed set).


Introduction
A two place function M is called a quasi-arithmetic mean (see e.g. [1]) if it can be written in the form where I is a non-empty, open interval, ϕ : I → R is a continuous, strictly monotonic function. This class is a close relative of quasi-sums (see e.g. [19]) and it is in the center of research of several authors (see e.g. [8] and the references therein). It contains classical well-known means like the arithmetic mean, geometric mean, harmonic mean and so on. Currently the operator version is also defined and investigated (see e.g. [10]).

An Extension Theorem for Conditionally Additive Functions
Let D ⊂ R be a subset, and D Δ := {(x, y) ∈ R 2 | x, y, x + y ∈ D }.
A function g : D → R is called conditionally additive on D (see [7]) if g(x) + g(y) = g(x + y), Proof. Let d ∈ diadD, then x = n i=1 δ i d i and for and arbitrary d i ∈ D there exists j i ∈ N, such that sign(d i ) 1 2 .
. The other direction of the inclusion is trivial.
Proof of Theorem 1. If D has a single element, the statement is trivial. We can assume without losses that D has at least two elements, and at least one is positive, say 0 < r ∈ D. Because of the previous lemma, it is enough to prove the statement for D r : For an arbitrary x ∈ diadD r there is a unique integer n x ∈ Z such that Because of its definition x ∈ D r . Assume that g : D r → R is conditionally additive, and let's define Firstly, we prove that a is an extension of g. For this we distinguish two cases: . This entails a is really an extension of g.
Secondly, we prove that a is additive on diadD r = diadD. Let x, y ∈ diadD r be arbitrary elements, and y= n y r 2 + y . We distinguish two cases again: = n x g r 2 + m y g r 2 + g(x ) + g(y ) = a(x) + a(y). That is to say, a is additive on the required set. Thirdly, we extend a onto the whole real line. It is clear that diadD with the usual addition is a subsemigroup of the additive group of real numbers. We can apply the theorem of Dhombres and Ger (see [9] or [11,Theorem 18.1.1.]). Which exactly says that there is a (not necessarily unique) homomorphism

Properties of Quasi-Arithmetic Expressions
Let I ⊂ R be a proper interval, ϕ : I → R be an invertible function such that its inverse has a dyadically closed domain. Then the following two place function is called a quasi-arithmetic expression generated by ϕ.
Besides of the required conditions, A ϕ is well-defined.
If ϕ continuous and strictly monotone, then it fulfils the requirements above. The resulted set is the class of quasi-arithmetic means.
If ϕ is an invertible additive function, then the corresponding quasiarithmetic expression is the arithmetic mean. If ϕ = a • log, where a is an invertible additive function again, we get the geometric mean. These phenomenon are called absorbing irregularity. In other words, a very regular expression can be generated by a very irregular one.
The following two observations shows that irregularity of the generating function is not always absorbed. As a consequence we have that the class of quasi-arithmetic expressions is really larger than the class of quasi-arithmetic means. Proof. Let I = R and a : R → R be an arbitrary non-continuous, additive, self-bijection of the reals, and ϕ = exp •a, then the quasi arithmetic expression generated by ϕ is the following We prove that A ϕ is non-continuous, so it cannot be a quasi-arithmetic mean. Assume that A ϕ is continuous, then the image of every connected set by A ϕ is connected. In all cases we get the boundedness of a either from below or from above. It follows that a is continuous, which is a contradiction.
The quasi-arithmetic means have the intern property, that is, their values are always between the minimum and the maximum of the variables. Intern property of the construction in the previous proof is an open problem, however, it is possible to construct a quasi-arithmetic expression which is non-continuous and it is not a mean.

Observation 2. There is a non-continuous quasi-arithmetic expression which is not a mean.
Proof. 1 Let ϕ : R → R be the following function So, A ϕ is not continuous.

Equality Problem of Quasi-Arithmetic Expressions
Here we solve the following problem. Let ϕ, ψ : I → R be invertible functions with inverses having dyadically closed domains. What is the sufficient and necessary condition of the equality of the generated quasi-arithmetic expressions? That is to say, solve the following functional equation!
1 Gyula Maksa's example, Oral communication. Proof. From the assumption A ϕ = A ψ we have x,y∈ I.
We can assume without losses that 0 ∈ D. Substituting v = 0 and b := f (0) into the equation above we have Using (3) and (4) we get Let g : D → R, g(u) := f (u) − b, then the previous equation entails the conditional additivity of g, that is to say, Applying the extension theorem (Theorem 1.) for g, we have that there is and additive function a : R → R such that a |D = g. Using this and the definitions of the functions f and g, we have This exactly means that So, the proof is ready.

Open Problems
Characterization of quasi-arithmetic means is well-known. (v) Symmetry: M (x, y) = M (y, x); The next theorem due to Aczél can be found in [2].
Conditions (iii)-(v) are fulfilled by quasi-arithmetic expressions too. The conditions (i) and (iv) implies intern property, that is, the value of the function is between the minimum and the maximum of the variables.
These ideas are motivate the following open problems: Open problem 1. Is it possible to construct a non-continuous quasi-arithmetic expression which is a mean?
Open problem 2. Is it true that (iii)-(v) (maybe with additional conditions different from (i) and (ii)) characterize quasi-arithmetic expressions and (i) with (iii)-(v) characterize quasi-arithmetic expressions with intern property?
If the answer is affirmative for the first, so is for the second.