2022/05/09 by Cédric de Lacroix, de Lacroix, Cédric, Luigi Santocanale +1
Mathematics · #Algebra over a field #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Computer science #Distributive property #FOS: Computer and information sciences #FOS: Mathematics #Hilbert space #Homotopy and Cohomology in Algebraic Topology #Lattice (music) #Logic (math.LO) #Logic in Computer Science (cs.LO) #Mathematics #Morphism #Negation #Pure mathematics
paper · doi:10.48550/arxiv.2205.04111
openalex publication_date 2022/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney's notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices Mn and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.