Existence of monoids compatible with a family of mappings

The set-up we work with here consists of non-empty sets S and X, a mapping f : S × X → X and an element x0 ∈ X. The mapping f will also be regarded as a family of mappings {fs}s∈S, where fs : X → X is given by fs(x) = f(s, x) for all x ∈ X. A subset X ′ of X is said to be f -invariant if f(S ×X ) ⊂ X , i.e., if fs(X ) ⊂ X ′ for each s ∈ S, and the mapping f is said to be minimal if the only f -invariant subset of X containing x0 is X itself. Of course, the definition of being minimal depends on x0, and so it would perhaps be better to denote this property as being x0-minimal or something similar. However, we consider the element x0 to be fixed and tend not to bring it into the notation. The basic assumption made in most of the results below is that the mapping f is minimal.