Branching inset entropies on open domains

Summary

The forms of all semisymmetric, branching, multidimensional measures of inset information on open domains are determined. This is done for both the complete and the possibly incomplete partition cases. The key to these results is to find the general solution of the functional equation

$$(*)\Delta \left( {\begin{array}{*{20}c} {E,F} \\ {p,q} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup F,G} \\ {p + q,r} \\ \end{array} } \right) = \Delta \left( {\begin{array}{*{20}c} {E,G} \\ {p,r} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup G,F} \\ {p + r,q} \\ \end{array} } \right)$$

for all(p, q, r) ∈ D ₃ :={(p ₁ , p ₂ , p ₃ )∣p _i ∈]0, 1[ ^k (i = 1,2,3),p ₁ + p ₂ + p ₃ ∈]0, 1[ ^k } and all (E, F, G) for which there exists anH with (E, F, G, H) ∈ ℒ₄:={(E ₁ ,⋯,E ₄ )∣ E _i ∈ ℛ\{Ø},E ₁∩E _j = Ø for alli ≠ j}. Here, ℛ is a ring of subsets of a given universal set. Functional equation (*) is solved by use of the general solution of the generalized characteristic equation of branching information measures.