A Criterion for Mapping Homogeneity

Abstract

The paper proposes a definition of a homogeneous mapping based on the concept of a group action. This definition can be used to describe generalizations of the concept of "homogeneous function," including positive, absolute, limited homogeneity, lambda-homogeneity, and homogeneous distributions. The main result is as follows: if a group acts on the set of assignments and values of the mapping, and acts commutatively on the set of values, then the mapping is homogeneous with respect to these actions if and only if the homogeneity condition is satisfied for the generating set of this group.
 In most cases, the assignment sets and value sets of mappings are vector spaces, and the multiplicative group of the main field or its subgroup acts as an acting group. Therefore, it is important to know their generating sets. For example, for the multiplicative group R+ of positive real numbers, these are numerical intervals, possibly with excluded null sets. Thus, it follows that the homogeneity of a function, understood traditionally, is guaranteed by the fulfillment of the homogeneity condition for any numerical interval from R+. This fact was previously only established for differentiable functions using the well-known Euler identity for homogeneous functions.