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Horn sentences in identity theory1

Published online by Cambridge University Press:  12 March 2014

K. I. Appel
K. I. Appel*
Affiliation:
University of Michigan
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Extract

Horn [2] obtained a sufficient condition for an elementary class to be closed under direct product. Chang and Morel [1] showed that this is not a necessary condition. We will show that, if consideration is restricted to identity theory, that is, a first-order predicate calculus with equality but no other relation symbols or operation symbols, Horn's condition is necessary and sufficient.

A model for identity theory consists of a non-empty domain A, but no relations or operations except equality. If I is an index set, and for each is a model for identity theory, then the direct product of the is a model for identity theory and has domain A, the cartesian product of the Ai.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 4 , December 1959 , pp. 306 - 310
Copyright
Copyright © Association for Symbolic Logic 1952

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Footnotes

1

This paper is a portion of a dissertation submitted to the graduate school of the University of Michigan in partial fulfillment of the requirements for the Ph.D. degree. The author wishes to thank Professor R. C. Lyndon for his guidance and encouragement and Professor John Addison for his many suggestions.

References

[1]Chang, C. C. and Morel, A., On closure under direct product, this Journal, vol. 23 (1958), pp. 149–154.Google Scholar
[2]Horn, A., On sentences which are true of direct unions, this Journal, vol. 16 (1951), pp. 14–21.Google Scholar