Bend sets, \(N\)-sequences, and mappings (Q1777817)

Let a dendroid \(X\) be of type \(N\) between points \(p\) and \(q\), with sequences \(\{p_n\}\), \(\{p_n'\}\), \(\{p_n''\}\), \(\{q_n\}\), \(\{q_n'\}\), \(\{q_n''\}\) (i.e. \(\{p_n''\} \in q_nq_n'\setminus\{q_n, q_n'\}\) and \(\{q_n''\} \in p_np_n'\setminus\{p_n, p_n'\}\) such that \(pq =\) Lim\(p_np_n' =\) Lim\( q_nq_n'\) and \(p=\lim p_n=\lim p_n'=\lim p_n''\) and \(q=\lim q_n=\lim q_n'=\lim q_n''\)). A surjective mapping \(g:X \to Y\) from \(X\) onto a dendroid \(Y\) is said to be \textit{admissible} if Ls\([g(p_nq_n'')\cap g(q_n''p_n')]\cap \) Ls\([g(q_np_n'')\cap g(p_n''q_n')]=\emptyset \). This concept generalizes the notion of a triade \((X,g,Y)\) having property (*) from \textit{S. T. Czuba}'s paper [General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 121--123 (1988; Zbl 0648.54031)]. The main result is as follows. Let an admissible mapping \(g:X \to Y\) between dendroids \(X\) and \(Y\) be given. Then \(Y\) is neither smooth, nor contractible, nor selectible, and it admits no associative mean.