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Journal: Annales Mathematicae Silesianae

The alienation phenomenon and associative rational operations

Roman Ger

Language: EN | Published: 30-09-2013 | Abstract | pp. 75-88

The alienation phenomenon of ring homomorphisms may briefly be described as follows: under some reasonable assumptions, a map f between two rings satisfies the functional equation
(*)       f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y)
if and only if f is both additive and multiplicative. Although this fact is surprising for itself it turns out that that kind of alienation has also deeper roots. Namely, observe that the right hand side of equation (*) is of the form Q(f(x), f(y)) with the map Q(u, v) = u+v+uv being a special rational associative operation. This gives rise to the following question: given an abstract rational associative operation Q does the equation
f(x + y) + f(xy) = Q(f(x), f(y))
force f to be a ring homomorphism (with the target ring being a field)?
Plainly, in general, that is not the case. Nevertheless, the 2-homogeneity of f happens to be a necessary and sufficient condition for that effect provided that the range of f is large enough.

2013-09-30



Journal: Annales Mathematicae Silesianae

Alienation of the Jensen, Cauchy and d’Alembert equations

Barbara Sobek

Language: EN | Published: 23-09-2016 | Abstract | pp. 181-191

Let (S,+) be a commutative semigroup, σ: S→S be an endomorphism with σ² = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f,g: S→K of the functional equation
f(x+y) + f(x+σ(y)) + g(x+y) = 2f(x) + g(x)g(y)   for x,y∈S.
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

2016-09-23



Journal: Annales Mathematicae Silesianae

Alienation of Drygas' and Cauchy's functional equations

Youssef Aissi , Driss Zeglami , Brahim Fadli

Language: EN | Published: 13-04-2021 | Abstract | pp. 131-148

Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation
f(x + y) + g(x + y) + g(x - y) = f(x)f(y) + 2g(x) + g(y) + g(-y).
We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.

2021-04-13



Journal: Annales Mathematicae Silesianae

On the alienation of multiplicative and additive functions

Mohamed Chakiri , Abdellatif Chahbi , Elhoucien Elqorachi

Language: EN | Published: 15-11-2024 | Abstract

Given S a semigroup. We study two Pexider-type functional equations
f(xy) + g(xy) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) + ∫_Sg(xyt)dμ(t) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
for unknown functions f and g mapping S into ℂ, where μ is a linear combination of Dirac measures (δ_{z_i}})_i∈I for some fixed elements (z_i)_i∈I contained in S such that ∫_Sdμ(t) = 1.
The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations
f(xy) = f(x) + f(y),  g(xy) = g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) = f(x) + f(y),  ∫_Sg(xyt)dμ(t) = g(x)g(y),   x,y ∈ S,
respectively.

2024-11-15