Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphisms

On the general solution of a generalization of the Goła̧b–Schinzel equation

“…If conditions (A1)-(A3) hold and the function X \ f −1 ({0}) ∋ x → f (x) ∈ K is injective, then, under some additional assumptions, we get a useful description of the general solution of equation (1). It has been applied in [16].…”

“…By (11), we see that w(c) ∈ T , so (12) yields c = e. This means that a = b. It remains to prove that f, M, • satisfy equation (1). Hence fix x, y ∈ X.…”

The Gołab–Schinzel equation and its generalizations

“…Some further applications of (GS) have been given by -P. Javor ([45], [46]): associative operations; -J. Aczél and S. Go lab ( [4], cf. [3], p. 311-315): subsemigroups of the semigroup of the affine mappings of R (t → αt + β); -P. Plaumann and S. Strambach ( [71]): classification of quasialgebras;…”

“…equation (4). Moreover in [23] it is proved that for n > 0 there is a strict connection between solutions f, g : X → K of the two functional equations…”

On solutions of a generalization of the Gołab–Schinzel equation

“…f (x + f (x) k y) = f (x)f (y) (2) where the set {x ∈ X : f (x) = 0} has an algebraically interior point, then there exists a linear functional g : X → R such that (i) in the case where k is an odd number,…”