Solution of generalized bisymmetry type equations without surjectivity assumptions (Q1291178)

The \(m\times n\) generalized bisymmetry equation (all functions unknown) \[ F[G_{1}(x_{11},\ldots,x_{1n}),\ldots, G_{m}(x_{m1},\ldots,x_{mn})]= G[F_{1}(x_{11},\ldots,x_{m1}), \ldots,F_{n}(x_{1n},\ldots,x_{mn})] \] characterizes consistent aggregation in mathematical economics. It has a long history [see e.g. the surveys \textit{J. van Daal} and \textit{A. H. Q. M. Merkies} in Measurement in Economics, ed. W. Eichhorn, Physica, Heidelberg, 607-637 (1987) and \textit{J. Aczél} in Choice, Decision, and Measurement, ed. A. A. J. Marley, Lawrence Erlbaum Ass. Publ., Mahwah, NJ, 225-233 (1997)]. Recent results assume ``surjectivity'' conditions of the following type (not spelled out in this paper): holding all but one variable constant the (unknown) function still assumes all values it would without fixing any of the variables. The present paper disposes of such conditions on Cartesian product of nondegenerate real intervals, while assuming continuity and strict monotonicity in each variable. The proof is difficult. It not only applies the usual double induction but has to piece together solutions on subintervals. However, the usual result is obtained: all unknown functions are quasi-sums (i.e. continuous isotopes of sums).