In this work, we investigate various combinatorial properties of Borel ideals on countable sets. We extend a theorem presented in [13] and identify an $F_\sigma $ tall ideal in which player II has a winning strategy in the Cut and Choose Game, thereby addressing a question posed by J. Zapletal. Additionally, we explore the Ramsey properties of ideals, demonstrating that the random graph ideal is critical for the Ramsey property when considering more than two colors. The previously known result for two colors is extended to any finite number of colors. Furthermore, we comment on the Solecki ideal and identify an $F_\sigma $ tall K-uniform ideal that is not equivalent to $\mathcal {ED}_{\text {fin}}$, thereby addressing a question from M. Hrušák’s work [10].

The algebraic counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed ω-semigroups of width 2 and height ωω . This paper completes the description of this algebraic hierarchy. We first give a purely algebraic decidability procedure of this partial ordering by introducing a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees.The Wagner degree of any ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every degree of this hierarchy.

The algebraic study of formal languages shows that ω-rational sets correspond precisely to the ω-languages recognizable by finite ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the 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 by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height ωω .