The model-theoretic significance of complemented existential formulas

Extract

Let T be an inductive, first-order theory in a language L, let E(L) denote the set of existential L-formulas, and let E(T) denote the distributive lattice of equivalence-classes φ^T of formulas φ ∈ E(L) with respect to equivalence in T. We consider three types of ‘complements’ in E(T): Let φ^T, ψ^T ∈ E(T) and suppose φ^T ∏ ψT = 0. Then ψ^T is a complement of φ^T, if φ^T ∐ ψ^T = 1; ψ^T is a pseudo-complement of φ^T, if for all μ^T ∈ E(T), (φ^T ∐ ψ^T) = 0 implies μ^T ≤ ψ^T; ψ^T is a “weak complement of φ^T, if for all μ^T ∈ E(T), (φ^T ⋰ ψ^T) ∐ μ^T = 0 implies μ^T = 0. The following facts are obvious: A complement of φ^T is also a pseudo-complement of φ^T and a pseudo-complement of φ^T is also a weak complement of φ^T. Any φ^T has at most one pseudo-complement; it is denoted by φ^T*. The relations ‘ψ^T is the complement of φ^T’ and ‘ψ^T is a weak complement of φ^T’ are symmetrical. We call φ^T (weakly, pseudo-) complemented if φ^T has a (weak, pseudo-) complement, and we call E(T) (weakly, pseudo-) complemented if every φ^T is (weakly, pseudo-) complemented.

The object of this note is to characterize (weakly, pseudo-) complemented existential formulas in model-theoretic terms, and conversely to characterize some classical notions of Robinson style model theory in terms of these formulas. The following theorems illustrate the second approach.