Cuts and completions Algebraic aspects of structural proof theory

In this thesis we look at different aspects of the interplay between structural proof theory and algebraic semantics for several non-classical propositional logics. Concretely, we explore connections between proof theory and algebra as they relate to structural sequent and hypersequent calculi for intermediate and substructural logics. Such connections are particularly strong for logics associated with the levels $\mathcal{P}_3$ and $\mathcal{N}_2$ of the substructural hierarchy introduced by Ciabattoni, Galatos, and Terui. Therefore, we investigate different algebraic aspects of these two levels. Among the algebraic aspects considered, completions of lattices and lattice-based algebras take on a prominent role.