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Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ℒ = ⟨F, R, ρ⟩ be a system language. Given a class of ℒ-systems K and an ℒ-algebraic system A = ⟨SEN,⟨N,F⟩⟩, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ⟨ SEN,⟨N,F⟩,R 𝔄 ⟩, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ℒ-morphism ⟨ F,α⟩ : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ℒ-algebraic system A, provided that K is closed under subsystems and reduced products.