Sharp and Unsharp Structures. A Unifying Framework for Algebraic Logic

This work lies within the realm of algebraic logic, focusing on structures that are central in their respective fields and extensively studied. In the first part, it investigates the categorical equivalence between (indexed) Boolean algebras and regular double Stone algebras. Despite their strong categorical relation, distinct model-theoretic aspects are identified. However, their structural theories demonstrate a cohesive treatment, preserving significant elementary properties such as finite categoricity. Additionally, pivotal classes in algebraic logic, including Boolean algebras, Stone algebras, Kleene algebras, and 3-valued MV algebras, find ample representation through injective hulls in regular double Stone algebras. In the second part, the focus shifts to structures relevant in the context of quantum logics. Starting from the concept of a block in an orthomodular lattice, the aim is to capture a smooth generalization of the theory of orthomodular lattices in the case of non-orthocomplemented lattices. Under certain conditions, the theory of orthomodular lattices seamlessly extends, elucidating the natural transition in the presence of smooth conditions.