Homological methods in the solution of certain functional equations (Q1821974)

The author investigates the solution \(f: S^ n\to A\) of the following two functional equations: \[ \lambda (x_ 1)f(x_ 2,...,x_{n+1})+\sum^{m}_{k=1}(-1)^ k f(x_ 1,...,x_{k-1},x_ kx_{k+1},x_{k+2},...,x_{n+1})+ \] \[ +(-1)^{n+1} f(x_ 1,...,x_ n)=0 \] \(\sum^{n}_{k=1}(-1)^ k f(x_ 1,...,x_{k- 1},\epsilon,x_ k,...,x_{n-1})=0\) where S is a monoid, R is a ring with identity, A is a left unitary R-module, \(\lambda\) : \(S\to R^*\) is a semigroup homomorphism \((R^*\) is R regarded as a semigroup under multiplication), \(\epsilon\) is a fixed element of Cen S. The solutions f are expressed (under some algebraic conditions) by functions \(g:S^{n- 1}\to A\) and \(g:S^{n+1}\to A\). The expressions have the same form for g as the left sides of the equations for f.