Decomposer and associative functional equations (Q935929)

The following functional equations are considered. Associative equations: \[ f(xf(yz))=f(f(xy)z),\qquad f(xf(yz))=f(f(xy)z)=f(xyz). \] Decomposer equations: \[ f(f^*(x)f(y))=f(y),\qquad f(f(x)f_*(y))=f(x). \] Strong decomposer equations: \[ f(f^*(x)y)=f(y),\qquad f(xf_*(y))=f(x). \] Canceler equations: \[ f(f(x)y)=f(xy),\qquad f(xf(y))=f(xy),\qquad f(xf(y)z)=f(xyz), \] where \(f^*,f_*\) are (supposed to be unique) functions with \(f^*(x)f(x)=f(x)f_*(x)=x\,\,(x\in X).\) The function \(f:X\to X\) is unknown, \(X\) is a set with a binary operation ``\(\cdot\)''. In some cases \(X\) is a semigroup or a group. The authors solve these equations under various assumptions, study relations and equivalence between them. It is also proved that the associative equations and the system of strong decomposer and canceler equations do not have nontrivial solutions in the simple groups.