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Partial orderings of fixed finite dimension: Model companions and density1

Published online by Cambridge University Press:  12 March 2014

Alfred B. Manaster  and
Jeffrey B. Remmel
Alfred B. Manaster
Affiliation:
University of California, San Diego, La Jolla, California 92093
Jeffrey B. Remmel
Affiliation:
University of California, San Diego, La Jolla, California 92093
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Extract

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.

In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 4 , December 1981 , pp. 789 - 802
Copyright
Copyright © Association for Symbolic Logic 1981

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Footnotes

1

We are indebted to J. Schmerl for introducing the notion of model companion to us, providing the lemma of §3, and realizing that that lemma provides a crucial step in generalizing our results about 2-dimensional partial orderings to the results in §3.

References

REFERENCES

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