The generalized equation of bisymmetry: Solutions based on cancellative abelian monoids (Q1296962)

The functional equation of \(m\times n\) generalised bisymmetry is \[ G\bigl(F_1 (x_{11}, \dots, x_{1n}), \dots, F_m(x_{m1}, \dots, x_{mn})\bigr)= F\bigl(G _1(x_{11}, \dots, x_{m1}), \dots, G_n(x_{1n}, \dots, x_{mn})\bigr)\tag{1} \] where \[ \begin{aligned} F_i: & X_{i1} \times\cdots \times X_{in}\to Y_i;G: X_{1j} \times\cdots \times X_{mj}\to Z_j\\ F: & Z_1\times \cdots\times Z_n\to S;G:Y_1 \times\cdots \times Y_m\to S.\;(i=1, \dots, m;j=1,\dots,n). \end{aligned} \] The equation (1) is also known as the aggregation equation. This alternate name arose from a model which links microeconomics and macroeconomics. If \((T,+)\) is an abelian semigroup with \(T\subset S\) then the equation (1) is satisfied by the functions given by \[ \begin{aligned} & F(z_1,\dots, z_n)=h_1(z_1)+ \cdots+ h_n(z_n)\\ & G(y_1, \dots, y_m)=g_1(y_1) +\cdots +g_m(y_m)\\ & F_i(x_{i 1}, \dots,x_{in}) =g_i^{-1}\bigl(f_{i1} (x_{i1}+ \cdots+ f_{in}(x_{in}) \bigr)\\ & G_j(x_{1j}, \dots, x_{mj})= h_j^{-1}\bigl(f_{1j}(x_{1j}) +\cdots+ f_{mj}(x_{m j}) \bigr)\end{aligned} \] where \(f_{ij}: X_{ij}\to T\), \(g_i:Y_i\to T\), \(h_j:Z_j\to T\), \(x_{ij}\in Xij\) and \(g_i\), \(h_j\) are bijections. The author refers to the formal set of functions so defined as the protocol for the equation (1). \textit{J. Aczel}, \textit{G. Maksa} and \textit{M. A. Taylor} [J. Math. Anal. Appl. 214, No. 1, 22-35 (1997)] have provided characterisations of solutions to the equation (1) which satisfy the protocol with \((T,+)\) an abelian group. Unfortunately the abelian group constraint is too strong to accommodate those simple aggregation models which involve the straightforward addition of monetary values. In this paper the abelian group replaced by a cancellative abelian monoid and the author gives a necessary and sufficient condition for the solutions to the equation (1) to be described in terms of a single cancellative abelian monoid.