The notion of independence in categories of algebraic structures, part I: Basic properties

AbstractWe define a formula φ(x;t) in a first-order language L, to be an equation in a category of L-structures K if for any H in K, and set p = {φ(x;a1);i ϵI, ai ϵ H} there is a finite set I0⊂I such that for any f:H→ F in K, .We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every quantifier-free formula is equivalent in T to a boolean combination of (quantifier-free) equations in Mod(T), the category of models of T with embeddings for morphisms.Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability (and actually identical to it in the case of equational theories) but which, in our context, has an algebraic character