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Generalized prime models

  • Robert Fittler (a1)

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A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

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[1]Chen, C. C. and Grätzer, G., On the construction of complemented lattices, Journal of Algebra, vol. 11 (1969), pp. 5663.
[2]Fittler, R., Categories of models with initial objects, Category theory, homology theory and their application. I, Springer Lecture Notes, vol. 86, 1969.
[3]Grätzer, G., On the existence of free structures over universal classes, Mathematische Nachrichten, vol. 36 (1968), pp. 135140.
[4]Grätzer, G., Free Σ-structures, Transactions of the American Mathematical Society, vol. 135 (1969), pp. 517542.
[5]Grätzer, G., Universal algebra, Van Nostrana, Princeton, N J., 1968.
[6]Vaught, R. L., Models of complete theories, Bulletin of the American Matematical Society, vol. 69 (1963), pp. 299313.
[7]Vaught, R. L., Denumerable models of complete theories, Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics (Warsaw, 1959), Warszawa, 1961, pp. 303321.

Generalized prime models

  • Robert Fittler (a1)

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