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

Abstract

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.

Key Words:

• κ-Filtered direct product
• κ-Filtered intersection
• Lyndon classes
• Quasivarieties
• Subdirect product
• Upward κ-directed poset
• Union of ⟨κ
• ⊑⟩-directed system

2000 Mathematics Subject Classification:

• Primary 03G99, 18C15
• Secondary 68N30

ACKNOWLEDGMENTS

As always, I feel the need to acknowledge the influence that the pioneering works of Wim Blok and Don Pigozzi, Janusz Czelakowski, and Josep Maria Font and Ramon Jansana have had on mine. More specifically, for the developments reported on in this article, I followed work of Czelakowski and Elgueta, that was based on preceding work of Elgueta on the model theory of equality-free first-order logic in the context of abstract algebraic logic.

Warm thanks to Don Pigozzi and to Roger Maddux for some discussions that helped clarify some aspects of Czelakowski and Elgueta (1999). Thanks go also to Charles Wells and Giora Slutzki for guidance and support.

Finally, I would like to thank an anonymous referee for “Communications in Algebra” for several useful comments and suggestions.

Notes

Communicated by I. Swanson.