A GAME THEORETICAL APPROACHTO THE ALGEBRAIC COUNTERPARTOF THE WAGNER HIERARCHY: PART I

The algebraic study of formal languages shows that ω-rational sets correspond precisely to the ω-languages recognizable byfinite ω-semigroups. Within this framework, we provide a constructionof the algebraic counterpart of the Wagner hierarchy. We adopta hierarchical game approach, by translating the Wadge theory fromthe ω-rational language to the ω-semigroup context. More precisely,we first show that the Wagner degree is indeed a syntactic invariant.We then define a reduction relation on finite pointed ω-semigroups bymeans of a Wadge-like infinite two-player game. The collection of thesealgebraic structures ordered by this reduction is then proven to be isomorphicto the Wagner hierarchy, namely a well-founded and decidablepartial ordering of width 2 and height ωω