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Boolean Algebra Retracts

Published online by Cambridge University Press:  20 November 2018

Timothy Cramer
Timothy Cramer*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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A Boolean algebra B is a retract of an algebra A if there exist homomorphisms ƒ: B → A and g: A → B such that gƒ is the identity map B. Some important properties of retracts of Boolean algebras are stated in [3, §§ 30, 31, 32]. If A and B are a-complete, and A is α-generated by B, Dwinger [1, p. 145, Theorem 2.4] proved necessary and sufficient conditions for the existence of an α-homomorphism g: A → B such that g is the identity map on B. Note that if a is not an infinite cardinal, B must be equal to A. The dual problem was treated by Wright [6]; he assumed that A and B are σ-algebras, and that g: A → B is a σ-homomorphism, and gave conditions for the existence of a homomorphism ƒ:B → A such that gƒ is the identity map.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 339 - 344
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dwinger, Ph., Retracts in Boolean algebras, Proc. Sympos. Pure Math., Vol. II, pp. 141–151 (Amer. Math. Soc, Providence, R. I., 1961).Google Scholar
2. Dwinger, Ph., Introduction to Boolean algebras, Hamburger Mathematische Einzelschriften, Heft 40 (Physica-Verlag, Wiirzburg, 1961).Google Scholar
3. Halmos, P. R., Lectures on Boolean algebras (Van Nostrand, Princeton-Toronto-New York-London, 1963).Google Scholar
4. Luxemburg, W. A. J., A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras, Fund. Math. 55 (1964), 239–247.Google Scholar
5. Sikorski, R., Boolean algebras, 2nd éd., Ergebnisse der Mathematik und ihrer Grenzgebiete, New Series, Band 25 (Springer-Verlag, Berlin-Göttingen-Heidelberg-New York, 1964).Google Scholar
6. Wright, J. D. M., A lifting theorem for Boolean a-algebras, Math. Z. 112 (1969), 326–334.Google Scholar