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Tensor products of polyadic algebras1

Published online by Cambridge University Press:  12 March 2014

Aubert Daigneault
Aubert Daigneault*
Affiliation:
Université De Montréal
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Extract

A basic concept of the theory of models is that of elementary equivalence of similar relational systems: two such systems are said to be elementarily equivalent if they satisfy the same first-order statements or, in other words, if they have the same (first-order) complete theory. It is possible to reformulate this notion of elementary equivalence of systems within the framework of algebraic logic by replacing theories by algebraic structures derived from them or more directly from the systems which are models of these theories. To any such theory T (or model of it), is indeed associated a locally finite polyadic algebra with equality, the underlying Boolean algebra of which is simply the well-known Tarski-Lindenbaum algebra of the theory. It is not hard to prove (see Section 6.1) that two systems are elementarily equivalent iff (i.e. if and only if) they have isomorphic polyadic. algebras. The possibility of replacing theories by algebraic structures and of reducing the purely logical concept of elementary equivalence to the algebraic one of isomorphism can be exploited to give a purely algebraic treatment of model-theoretic problems and suggests natural questions concerning these structures. The present paper illustrates that possibility.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 3 , September 1963 , pp. 177 - 200
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

The results of this paper were contained for the most part in Chapter III of the author's doctoral dissertation [4] (Princeton 1959). They were previously announced in [3]. The author is most grateful to the referee for his constructive criticism and his numerous suggestions.

References

[1]Bourbaki, N., Eléments de mathématiques, Livre II, Chapitre III, Algèbre multilinéaire, Hermann, Paris (1948).Google Scholar
[2]Church, A., Introduction to Mathematical Logic, Vol. 1, Princeton (1956), ix + 376 pp.Google Scholar
[3]Daigneault, A., Tensor products of polyadic algebras. Notices of the American Mathematical Society, 1959, p. 44.Google Scholar
[4]Daigneault, A., Products of polyadic algebras and of their representations. Doctoral dissertation, Princeton University (1959).Google Scholar
[5]Daigneault, A. and Monk, D., Representation theory for polyadic algebras, to appear in Fundamenta Mathematicae.Google Scholar
[6]Feferman, S. and Vaught, R. L., Bulletin of the American Mathematical Society 63 (1957), Abstracts 577 and 578.Google Scholar
[7]Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems. Fundamenta Mathematicae 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[8]Halmos, P. R., The basic concepts of algebraic logic, American Mathematical Monthly, Vol. 53 (1956), pp. 363–387.CrossRefGoogle Scholar
[9]Halmos, P. R., Algebraic Logic, I. Compositie Mathematica 12 (1955), pp. 217–249.Google Scholar
[10]Halmos, P. R., Algebraic Logic II. Fundamenta Mathematicae 43 (1957), pp. 255–325.Google Scholar
[11]Halmos, P. R., Algebraic Logic, III. Transactions of the American Mathematical Society 83 (1956) pp. 430–470.Google Scholar
[12]Halmos, P. R., Algebraic Logic, IV. Transactions of the American Mathematical Society 86 (1957) pp. 1–27.Google Scholar
[13]Halmos, P. R., Algebraic Logic, New York (Chelsea) (1962), 271 pp.Google Scholar
[14]Henkin, L., La structure algébrique des théories mathématiques, Paris (Gauthier-Villars) (1956), 52 pp.Google Scholar
[15]Henkin, L. and Tarski, A., Cylindric algebras, Proceedings of Symposia in Pure Mathematics, Vol. II, Lattice theory (1961), pp. 83–113.CrossRefGoogle Scholar
[16]Sikorski, R., Independent fields and cartesian products, Studia Mathematica 11 (1950) pp. 171–184.CrossRefGoogle Scholar
[17]Sikorski, R., Cartesian products of Boolean algebras. Fundamenta Mathematicae 37 (1950) pp. 25–54.CrossRefGoogle Scholar