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The model-theoretic significance of complemented existential formulas

Published online by Cambridge University Press:  12 March 2014

Volker Weispfenning
Volker Weispfenning*
Affiliation:
University of Heidelberg, 69 Heidelberg, Federal Republic of Germany
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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 φT of formulas φ ∈ E(L) with respect to equivalence in T. We consider three types of ‘complements’ in E(T): Let φT, ψT ∈ E(T) and suppose φT ∏ ψT = 0. Then ψT is a complement of φT, if φT ∐ ψT = 1; ψT is a pseudo-complement of φT, if for all μT ∈ E(T), (φT ∐ ψT) = 0 implies μT ≤ ψT; ψT is a “weak complement of φT, if for all μT ∈ E(T), (φT ⋰ ψT) ∐ μT = 0 implies μT = 0. The following facts are obvious: A complement of φT is also a pseudo-complement of φT and a pseudo-complement of φT is also a weak complement of φT. Any φT has at most one pseudo-complement; it is denoted by φT*. The relations ‘ψT is the complement of φT’ and ‘ψT is a weak complement of φT’ are symmetrical. We call φT (weakly, pseudo-) complemented if φT has a (weak, pseudo-) complement, and we call E(T) (weakly, pseudo-) complemented if every φT 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 4 , December 1981 , pp. 843 - 850
Copyright
Copyright © Association for Symbolic Logic 1981

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References

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