ISSN:
1420-8903
Keywords:
Keywords. Aggregation equation, generalized bisymmetry, injective.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000100050154
Permalink
