×

zbMATH — the first resource for mathematics

Axiomatizable classes with strong homomorphisms. (English) Zbl 0639.03035
The author offers as solution of Problem 1 in § 3.1 of A. I. Mal’tsev’s paper in Trud. Chetvert. Vsesoyuzn. Mat. S”ezda, Leningrad, 3-12 Iyulya 1961, 1, 169-198 (1963; Zbl 0191.295), the following theorem: All homomorphisms between members of an axiomatic class \({\mathcal K}\) are strong homomorphisms if and only if for each predicate there is an “S-axiom” true in all members of \({\mathcal K}\). For the intended definition of “S-axiom” the reader should look at the sufficiency part of the proof on page 118, rather than to the author’s definition on page 115. [The reviewer would have understood Mal’tsev’s problem to refer to all homomorphisms from members of \({\mathcal K}\) and not just to those whose target is also in \({\mathcal K}.]\)
Reviewer: G.Fuhrken
MSC:
03C52 Properties of classes of models
08C10 Axiomatic model classes
03C60 Model-theoretic algebra
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Andreka and I. Nemeti, Generalization of the concept of variety and quasi variety to partial algebras through category theory, Dissertationes Mathematicae, CCIV, p. 198. · Zbl 0518.08007
[2] H. Andreka and I. Nemeti, Formulas and ultraproduct in categories, Beitr?ge zur Algebra and Geometrie (8 (1979), pp. 133-151.
[3] Yu. L. Ershov, Solvability problems and constructive models, ?Nauka?, Moscow, 1980 (in Russian).
[4] H. Keisler and C. C. Chang, Model theory, Moscow, ?Mir?, 1977 (in Russian). · Zbl 0423.03041
[5] H. I. Keisler, Reduced products and Horn classes, Transactions of American Mathematical Society 117, pp. 307-328. · Zbl 0199.01103
[6] R. Lyndon, Existential Horn sentences, Proceedings of American Mathematical Society, pp. 994-998, · Zbl 0114.24902
[7] R. Lyndon, Properties preserved under homomorphisms, Pacific Journal of Mathematics 9, No. 1, pp. 143-154. · Zbl 0093.01101
[8] R. Lyndon, Properties preserved in subdirect products, Pacific Journal of Mathematics 9, No. 1, pp. 155-164. · Zbl 0093.01102
[9] J. ?o? and R. Suszko, On the infinite sums of models, Bulletin de l’Acad?mie Polonaise des Sci?nces 3, (1955), No. 4, p. 201-202.
[10] J. ?o? and R. Suszko, On the extending of models. II, Fundamenta Mathematicae 42, (1955), No. 2, p. 470.
[11] J. ?o?, Quelques remarques, theoremes et problemes sur les classes definissables d’algebres, In: Mathematical Interpretations of Formal System, Amsterdam, 1955.
[12] J. ?o?, On extending of models. I, Fundamenta Mathematicae 42, (1955), pp. 38-54. · Zbl 0065.00401
[13] A. I. Malcev, Some problems of the theory of classes of models, IV, All-Union Mathematical Congress, Leningrad, 1963, Trudy, t. 1, pp. 169-198 (in Russian).
[14] A. I. Malcev, Subdirect products of models, Doklady AN SSSR, 109, (1956), No. 2, pp. 264-266 (in Russian).
[15] A. I. Malcev, On classes of models with the generation operation, Doklady AN SSSR 116, (1957), No. 5, p. 738-741 (in Russian). · Zbl 0079.00703
[16] A. Marczewski, Sur les congruences et les proprictes positive d’algebres abstraites, Colloquium Mathematicum, 2, (1951), pp. 220-228.
[17] A. Robinson, Obstructions to arithmetical extension and the theorem of ?o? and Suszko, Indagationes Mathematicae, 21, (1939), No. 5, pp. 489-495. · Zbl 0173.00602
[18] A. Tarski, Contributions to the theory of models, III, Proc. Koninkl. nederl. acad. wet. A58, (1955), pp. 58-64. · Zbl 0058.24702
[19] S. Shelah, Uniqueness and characterization of prime models over sets for totally transcendental first-order theories, Journal of Symbolic Logic 37, pp. 107-113. · Zbl 0247.02047
[20] F. Golvin, Horn sentences, Annals of Mathematical Logic (1970), No. 4, pp. 389-422. · Zbl 0206.27801
[21] I. Gain, On classes of algebraic systems closed with respect to quotients, Universal Algebra and Applications, Banach Center Publications, Vol. 9, PWN-Polish Scientific Publishers, Warsaw, 1982, pp. 127-131.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.