Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T16:14:27.798Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of companionable, universal theories

Published online by Cambridge University Press:  12 March 2014

William H. Wheeler
William H. Wheeler*
Affiliation:
Indiana University, Bloomington, Indiana 47401
Get access Rights & Permissions [Opens in a new window]

Extract

A first-order theory is companionable if it is mutually model-consistent with a model-complete theory. The latter theory is then called a model-companion for the former theory. For example, the theory of formally real fields is a companionable theory; its model-companion is the theory of real closed fields. If a companionable, inductive theory has the amalgamation property, then its model-companion is actually a model-completion. For example, the theory of fields is a companionable, inductive theory with the amalgamation property; its model-completion is the theory of algebraically closed fields.

The goal of this paper is the characterization, by “algebraic” or “structural” properties, of the companionable, universal theories which satisfy a certain finiteness condition. A theory is companionable precisely when the theory consisting of its universal consequences is companionable. Both theories have the same model-companion if either has one. Accordingly, nought is lost by the restriction to universal theories. The finiteness condition, finite presentation decompositions, is an analogue for an arbitrary theory of the decomposition of a radical ideal in a Noetherian, commutative ring into a finite intersection of prime ideals for the theory of integral domains. The companionable theories with finite presentation decompositions are characterized by two properties: a coherence property for finitely generated submodels of finitely presented models and a homomorphism lifting property for homomorphisms from submodels of finitely presented models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 3 , September 1978 , pp. 402 - 429
Copyright
Copyright © Association for Symbolic Logic 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

[1]Atiyah, M. F. and MacDonald, I. G., Introduction to commutative algebra, Addison-Wesley, Reading, Massachusetts, 1969.Google Scholar
[2]Bacsich, P., Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis, vol. 5 (1975), pp. 45–55.CrossRefGoogle Scholar
[3]Bourbaki, N., Commutative algebra, elements of mathematics (English translation), Hermann, Paris, 1972.Google Scholar
[4]Carson, A. B., The model-completion of the theory of commutative regular rings, Journal of Algebra, vol. 27 (1973), pp. 136–146.CrossRefGoogle Scholar
[5]Cherlin, G., Algebraically closed commutative rings, this Journal, vol. 38 (1973), pp. 439–499.Google Scholar
[6]Comer, S., Complete and model-complete theories of monadic algebras, Colloquium Mathematicum, vol. 34 (1976), pp. 183–190.CrossRefGoogle Scholar
[7]Harary, F., Graph theory, Addison-Wesley, Reading, Massachusetts, 1969.CrossRefGoogle Scholar
[8]Hirschfeld, J. and Wheeler, W. H., Forcing, arithmetic, division rings, Lecture Notes in Mathematics, vol. 454, Springer-Verlag, Berlin and New York, 1975.CrossRefGoogle Scholar
[9]Kaplansky, I., Commutative rings (revised edition), University of Chicago Press, Chicago, 1974.Google Scholar
[10]Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, Massachusetts, 1966.Google Scholar
[11]Lipschitz, L. and Saracino, D., The model-companion of the theory of commutative rings without nilpotent elements, Proceedings of the American Mathematical Society, vol. 38 (1973), pp. 381–387.CrossRefGoogle Scholar
[12]Macintyre, A., Model-completeness for sheaves of structures, Fundamenta Mathematicae, vol. 81 (1973), pp. 73–89.CrossRefGoogle Scholar
[13]Ore, O., Theory of graphs, American Mathematical Society Colloquium Publications, vol. 38, American Mathematical Society, Providence, RI, 1962.Google Scholar
[14]Robinson, A., Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam, 1965.Google Scholar
[15]Sabidussi, G., Subdirect representations of graphs, Infinite and finite sets (Hajnal, A., Rado, R., Sós, V. T., Editors), vol. 3, pp. 1199–1226, Colloquia Mathematica Societatis Jdnos Bolyai, vol. 10, North-Holland, Amsterdam, 1975.Google Scholar
[16]Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
[17]Weispfenning, V., Model-completeness and elimination of quantifiers for subdirect products of structures, Journal of Algebra, vol. 36 (1975), pp. 252–277.CrossRefGoogle Scholar
[18]Wheeler, W. H., Model-companions and definability in existentially complete structures, Israel Journal of Mathematics, vol. 25 (1976), pp. 305–330.CrossRefGoogle Scholar