ω-Semigroups and the Fine Classification of Borel Subsets of Finite Ranks of the Cantor Space

The algebraic study of formal languages draws the equivalence between ω-regular languages and subsets of finite ω-semigroups. The ω-regular languages being the ones characterised by second order monadic formulas. Within this framework, in [1]we provide a characterisation of the algebraic counterpart of the Wagner hierarchy: a celebrated hierarchy of ω-regular languages. For this, we adopt a hierarchical game approach and translate, from the ω-regular language to the ω-semigroup context, the very few elements of the Wadge theory that the Wagner hierarchy is concerned with. We also define a reduction relation on subsets of finite ω-semigroups by means of a Wadge-like infinite two-player game, and then give a description of the resulting hierarchy of such subsets. Finally, we prove that this algebraic hierarchy is isomorphic to the Wagner hierarchy. We propose to extend this algebraic approach from the Wagner hierarchy to the first levels of the Borel hierarchy (of subsets of the Cantor Space) by considering infinite ω-semigroups – obtained as combinations of finite ω-semigroups – equipped with pseudoinverses.