Abstract. This is one in a series of papers studying measures of information in the so-called “mixed” theory of information (i.e. considering the events as well as their probabilities) on the “open” domain (i.e. without empty sets and zero probabilities). In this paper we find allβ-recursive, 3-semisymmetric inset entropies on the open domain. We do this by solving the fundamental equation of inset information of degreeβ (β∈ℝ) on the open domain.