On the equation of generalized bisymmetry with outer functions injective in each argument

Summary.

It is shown that provided F and G are injective in every argument, the functional equation of generalized $m \times n$ bisymmetry $(m,n \ge 2)$,¶¶$G(F_1(x_{11}, \hdots , x_{1n}),\hdots , F_m(x_{m1},\hdots, x_{mn}))$¶$= F(G_1(x_{11},\hdots , x_{m1}),\hdots , G_n(x_{1n},\hdots , x_{mn}))$¶may be reduced to ¶¶$G(\overline{F}_1(u_{11}, \hdots , u_{1n}),\hdots , \overline{F}_m(u_{m1},\hdots, u_{mn}))$¶$= F(\overline{G}_1(u_{11},\hdots , u_{m1}),\hdots ,\overline{G}_n(u_{1n},\hdots , u_{mn}))$¶where¶¶$F_i(x_{i1},\hdots , x_{in}) = \overline{F}_i (\varphi_{i1}(x_{i1}),\hdots , \varphi_{in}(x_{in})), G_j(x_{1j}, \hdots , x_{mj}) = \overline{G}_j(\varphi_{1j} (x_{1j}),\hdots, \varphi_{mj}(x_{mj}))$,¶¶\( \varphi_{ij} are surjections and $\overline{F}_i, \overline{G}_j$ are injective in every argument for all $1\le i \le m,\ 1\le j\le n$. The result is also shown to hold for a wider class of functional equations.