2024/06/21 by David Forsman, Forsman, David
Computer Science · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2406.15584
openalex publication_date 2024/06/21 · openalex created_date 2024/06/26 · openalex updated_date 2026/07/28
Eight categorical soundness and completeness theorems are established within the framework of algebraic theories. Exactly six of the eight deduction systems exhibit complete semantics within the cartesian monoidal category of sets. The multicategorical meta-theorem via soundness and completeness enables the transference of properties of families of models from the cartesian monoidal category of sets to Δ-multicategories C. A bijective correspondence R ↦ ΔR is made between context structures R and structure categories Δ, which are wide subcategories of FinOrd consisting of finite ordinals and functions. Given a multisorted signature σ with a context structure R, an equational deduction system \vdashR is constructed for R-theories. The models within ΔR-multicategories provide a natural semantic framework for the deduction system \vdashR for modelable context structures R. Each of the eight modelable context structures R is linked with a soundness and completeness theorem for the deduction system \vdashR.