2025/01/08 by Jan Paseka, Paseka, Jan, Thomas Vetterlein +1 · 1 citation
Decision Sciences · #06C15 #46C05 #81P10 #FOS: Mathematics #FOS: Physical sciences #Fuzzy and Soft Set Theory #Logic (math.LO) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2501.04482
openalex publication_date 2025/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
An orthoset is a non-empty set together with a symmetric and irreflexive binary relation ⊥, called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is orthogonal to every element. The collection of subspaces of a Hilbert space that are spanned by a single vector provides a motivating example. We say that a map f \colon X → Y between orthosets with 0 possesses the adjoint g \colon Y → X if, for any x ∈ X and y ∈ Y, f(x) ⊥ y if and only if x ⊥ g(y). We call f in this case adjointable. For instance, any bounded linear map between Hilbert spaces induces a map with this property. We discuss in this paper adjointability from several perspectives and we put a particular focus on maps preserving the orthogonality relation. We moreover investigate the category OS of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory iOS of irredundant orthosets with 0. iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of projective Hilbert spaces.