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Decidability and ℵ0-categoricity of theories of partially ordered sets

Published online by Cambridge University Press:  12 March 2014

James H. Schmerl
James H. Schmerl*
Affiliation:
University of Connecticut, Storrs, Connecticut 06268
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Abstract

This paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following.

Corollary 3.3. For every countable ℵ0-categoricalthere is a linear order of A such that (, <) is ℵ0-categorical.

Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory.

Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory.

There is a section dealing just with decidability of partially ordered sets, the main result of this section being

Theorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980 , pp. 585 - 611
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]Ash, C. J., Undecidable ℵ0-categorical theories, Notices of the American Mathematical Society, vol. 18 (1971), p. 423.Google Scholar
[2]Bacsich, P., The strong amalgamation property, Colloquium Mathtmaticum, vol. 33 (1975), pp. 13–23.CrossRefGoogle Scholar
[3]Baker, K. A., Fishburn, P. C. and Roberts, F. S., Partial orders of dimension 2, Networks, vol. 2 (1971), pp. 11–28.CrossRefGoogle Scholar
[4]Crossley, J. and Nerode, A., Combinatorial functors, Ergebnisse Mathematische Grenzgebiete, Band 81, Springer, New York, 1974.Google Scholar
[5]Dilworth, R. P., A decomposition theorem for partially ordered sets, Annals of Mathematics, vol. 51 (1950), pp. 161–166.CrossRefGoogle Scholar
[6]Dushnik, B. and Miller, E. W., Partially ordered sets, American Journal of Mathematics, vol. 63 (1941), pp. 600–610.CrossRefGoogle Scholar
[7]Ehrenfeucht, A., There are continuum ℵ0-categorical theories, Bulletin de l'Académie Polonaise des Sciences. Séries des Sciences Mathématiques, Astronomiques et Physiques, vol. 20 (1972), pp. 425–427.Google Scholar
[8]Eklof, P., Algebraic closure operators and strong amalgamation bases, Algebra Universalis, vol. 4 (1974), pp. 89–98.CrossRefGoogle Scholar
[9]Engeler, E., A characterization of theories with isomorphic denumerable models, Notices of the American Mathematical Society, vol. 6 (1959), p. 161.Google Scholar
[10]Fraïssé, R., Sur certaines relations qui généralisent l'ordre des nombres rationnels, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, vol. 237 (1953), pp. 540–542.Google Scholar
[11]Glassmire, W. Jr., There are 2ℵ0 countably categorical theories, Bulletin de l'Académie Polonaise des Sciences. Séries des Sciences Mathématiques, Astronomiques et Physiques, vol. 19 (1971), pp. 185–190.Google Scholar
[12]Grzegorczyk, A., Logical uniformity by decomposition and categoricity in ℵ0, Bulletin de l'Académie Polonaise des Sciences. Séries des Sciences Mathématiques, Astronomiques et Physiques, vol. 16 (1968), pp. 687–692.Google Scholar
[13]Henson, C. W., Countable homogeneous relational structures and ℵ0-categorical theories, this Journal, vol. 37 (1972), pp. 494–500.Google Scholar
[14]Herrmann, E., On Lindenbaum functions of ℵ0-categorical theories of finite similarity types, Bulletin de l'Académie Polonaise des Sciences. Séries des Sciences Mathématiques, Astronomiques et Physiques, vol. 24 (1976), pp. 17–21.Google Scholar
[15]Manaster, A. and Rosenstein, J., Two-dimensional partial orderings and recursion theory (preprint).Google Scholar
[16]Peretyatkin, M. G., On complete theories with a finite number of denumerable models, Algebra i Logika, vol. 12 (1973), pp. 550–576.Google Scholar
[17]Plotkin, J., Generic embeddings, Ph.D. thesis, Cornell University, 1968.Google Scholar
[18]Rabin, M. O., A simple method for undecidability proofs and some applications, Logic, Methodology and Philosophy of Science, Proceedings of the 1964 International Congress, North-Holland, Amsterdam, 1965, pp. 58–68.Google Scholar
[19]Rabin, M. O., Decidability of second-order theories and automata on finite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
[20]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[21]Rosenstein, J.G., ℵ0-categoricity of linear orderings, Fundamenta Mathematicae, vol. 44 (1969), pp. 1–5.CrossRefGoogle Scholar
[22]Ryll-Nardzewski, C., On the categoricity in power ≤ ℵ0, Bulletin de l'Académie Polonaise des Sciences. Séries des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1959), pp. 545–548.Google Scholar
[23]Schmerl, J.H., The decidability of some ℵ0-categorical theories, Colloquium Mathematicum, vol. 36 (1976), pp. 165–169.CrossRefGoogle Scholar
[24]Schmerl, J.H., On ℵ0-categoricity and the theory of trees, Fundamenta Mathematicae, vol. 94 (1977), pp. 121–128.CrossRefGoogle Scholar
[25]Schmerl, J.H., A decidable ℵ0-categorical theory with a non-recursive Ryll-Nardzewski function, Fundamenta Mathematicae, vol. 98 (1978), pp. 121–125.CrossRefGoogle Scholar
[26]Schmerl, J.H., ℵ0-categoricity of partially ordered sets of width 2, Proceedings of the American Mathematical Society, vol. 63 (1977), pp. 299–305.Google Scholar
[27]Svenonius, L., ℵ0-categoricity in first-order predicate calculus, Theoria (Lund), vol. 25 (1959), pp. 82–94.CrossRefGoogle Scholar
[28]Venning, M.C., Type structure of aleph-zero categorical theories, Ph.D. thesis, Cornell University, 1976.Google Scholar