The notion of independence in categories of algebraic structures, part III: equational classes

Abstract

Consider the following situation: K is an arbitrary class of structures in a fixed language L, H a structure in K, S a set of quantifier-free formulas in L with S=cl + (S), and Δ=cl(δ), where δ is a set of formulas in L such that every embedding H→F in K preserves δ, S⊂δ=cl + (δ). Given A⊂B, C⊂H, let us say that B is S-independent from C over A, if tp(B, C; H) is an S H -minimal extension of tp(B, A; H) where a type shall mean a Δ-type. We will show in this paper that if K is a universal class with AP, where every atomic formula is an equation, H is existentially closed, Δ=Qf, and S=cl + (At), then this independence relation satisfies all the natural properties that Shelah's one satisfies in stable theories. Infact we will show that S-independence is equivalent to R-independence where R is the set of all quantifier-free equations satisfying some other conditions