\(n\)-variable bisection.
Study of the equation \[ B[B(x_{11},\dots,x_{1n}),\dots,B(x_{n1},\dots,x_{nn})]=B[B(x_{11},...,x_{n1}),. ,\dots,B(x_{1n},\dots,x_{nn})] \] for real functions is considered to have started with papers by \textit{J. Aczél} [[1] Norske Vid. Selsk. Forh. (Trondheim) 19, No. 23, 83--86 (1946; Zbl 0030.02601); [2] Bull. Am. Math. Soc. 54, 392--400 (1948; Zbl 0030.02702)], who called it bisymmetry, even though for \(n=2\) it was subject of some algebraic studies earlier and also later under different names, like ``abelian'' (e.g. \textit{D.C. Murdoch} [Trans. Am. Math. Soc. 49, 392--409 (1941; Zbl 0025.10003; JFM 67.0082.01)], ``alternation'' (\textit{M. Sholander} [Bull. Am. Math. Soc. 55, 746--755 (1949; Zbl 0034.29802)], even ``entropic'' (\textit{I.M.H. Etherington}, Proc. Edinb. Math. Soc. (2) 13, 219--222 (1963; Zbl 0115.02201)]; ``etymologically `entropic' means `inner turning'\dots'' wrote the author), and ``medial'' (\textit{S. K. Stein}, Trans. Amer. Math. Soc. 85, 228--256 (1957; Zbl 0079.02402)]. [The reviewer tried later to switch the ``bisymmetric, bisymmetry'' terminology to ``medial, mediality'', see e.g. \textit{J. Aczél} and \textit{J. Dhombres}, Functional equations in several variables with applications to mathematics, information theory, and to the natural and social sciences, Chapter 17, Cambridge University Press (1989; Zbl 0685.39006) - to no avail]. In [1] it has been proved that, \(I\) being a nontrivial real interval, the function \(B: I^n\to I\) is a mean value (\(B(x,\dots,x)=x\) for all \(x\in I\)), bisymmetric, continuous, strictly increasing in each variable and symmetric if, and only if, there exists a continuous strictly monotonic function \(f: I \to \mathbb{R}\) such that \(B(x_1,\dots,x_n)=f^{-1}[\sum_{k=1}^n f(x_k)/n]\) for all \(x_k\in I\;(k=1,\dots,n)\). It has been stated in [2] that, if we drop the symmetry from among the above assumptions, then the general form of \(B\) will be \(B(x_1,\dots,x_n)=f^{-1}[\sum_{k=1}^n q_k f(x_k)],\) where \(q_1,\dots,q_n\) are positive constants, such that \(\sum_{k=1}^n q_k=1.\) The proof was given, however, only for \(n=2\) in order to ``simplify our considerations''. The first complete proof (by induction) in the nonsymmetric case for \(n>2\) appears in the paper under review. [An elegant noninductive proof has been given since by one of the authors of the present paper, Gy. Maksa, in Publ. Math. 61, No. 3--4, 663--669 (2002; Zbl 1017.39008)]. In the introduction the relevance of bisymmetry for measurement in mathematical psychology and psychophysics is argued. Also the generalized bisymmetry equation \[ C[B_1(x_{11},\dots,x_{1n}),\dots,B_m(x_{m1},\dots,x_{mn})] \] \[ =D[E_1(x_{11},\dots,x_{m1}),\dots,E_n(x_{1n},\dots,x_{mn})] \] is mentioned (mainly for \(m=n=2\)) and it is pointed out that the general continuous and strictly monotonic real solutions have not been satisfactorily determined (too strong surjectivity conditions). [Since then a general result without such blemish has been found, also by Gy. Maksa [Aequationes Math. 57, 50--74 (1999; Zbl. ....)].
scientific article; zbMATH DE number 1750031