Semi-homogeneous maps

Let V be a vector space over a field F. A functional f : V → F is called “homogeneous (of degree 1)” if f(λv) =λf(v) holds for all v ∈ V and all λ ∈ F, and “semi-homogeneous” if this equations only holds for all λ ∈ Im(f). If F = ℝ and Im(f) =ℝ⁺ = {x ∈ ℝ | x ≥ 0}, a semi-homogeneous map is usually called “positively homogeneous”. Certainly the most famous result in this area is Euler’s Homogeneous Function Theorem (Kudryavtsev (2002)): If f : ℝⁿ → ℝ is continuously differentiable then it is semi-homogeneous of degree 1 if and only if v ·∇f(v) =f(v) holds for all v ∈ ℝⁿ. Observe that a semi-homogeneous functional on V is automatically linear if dim(V )=1 and “often”linear in the other cases (cf. Fuchs et al. (1991) and Maxson and Van der Merwe (2002)). In this paper, we will completely characterize semi-homogeneous functionals. In fact, we will do this in a much more general context. That is, we will characterize semi-homogeneous maps from a set S to a certain monoid G acting on S. It will come as a pleasant surprise that this general view on semi-homogeneous maps gives information on two completely different topics, namely of algebraic structures called “planar nearrings” and of fixed point free monoid actions on sets. In a certain sense, semi-homogeneous maps, fixed point free monoid actions, and planar nearrings are basically the same.