Categorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems

The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems rather than universal algebras as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors rather than simply a system of relations on a single set. Congruence systems of structure systems are introduced and the Leibniz congruence system of a structure system is defined. Analogs of the Homomorphism, the Second Isomorphism and the Correspondence Theorems of Universal Algebra are provided in this more abstract context. These results generalize corresponding results of Elgueta for equality-free first-order logic. Finally, a version of Gödel’s Completeness Theorem is provided with reference to structure systems.