The multiplicative Cauchy equation on [0,1] and its application to the quasi-homogeneity equation

The (extended) homogeneity plays important, and often crucial role in economics, decision making and image processing. When studying the quasi-homogeneity equation, the multiplicative Cauchy equation on $[0,1]$ naturally appears. However, after a detailed analysis of its existing solutions it turned out that some solution of (extended) homogeneity equation was missing. The aim of the paper is to present a complete list of solutions of the multiplicative Cauchy equation and then to give the missing solution of quasi-homogeneity equation from the previous work. For a triangular norm T, we introduce a new concept of T-quasi-homogeneity and, for a strict t-norm T, we clarify the link between the quasi-homogeneity and the T-quasi-homogeneity.