Jean-Eric Pin introduced the structure of an ´ ω-semigroup in [PerPin04] as an algebraic counterpart to the concept of automaton reading infinite words. It has been well studied since, specially by Carton, Perrin [CarPer97] and [CarPer99], and Wilke. We introduce a reduction relation on subsets of ω-semigroups defined by way of an infinite two-player game. Both Wadge hierarchy and Wagner hierarchy become special cases of the hierarchy induced by this reduction relation. But on the other hand, set theoretical properties that occur naturally when studying these hierarchies, happen to have a decisive algebraic counterpart. A game theoretical characterization of basic algebraic concepts follows.