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A description of the projective Stone algebras

Published online by Cambridge University Press:  18 May 2009

Ivo Düntsch
Ivo Düntsch
Affiliation:
Freie Universitat Berlin, FB 10 Garystr. 21 1000 Berlin 33 W. Germany
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In his book [6] Grätzer sets a guideline for the research on Stone algebras; in view of his and Chen's triple characterization of Stone algebras [3], he considers a problem for these algebras solved if it can be reduced to a problem for Boolean algebras and one for distributive lattices with 1. In the same book he lists as Problem 53: describe the projective Stone algebras. For a Stone algebra L, let BL be its centre and DL its dense set. In order that L be projective, BL has to be a Boolean retract of BF for some free Stone algebra F, and DL has to be a retract of DF in the category of distributive lattices with 1. These conditions are, however, not sufficient, so one hopes to characterize the projective Stone algebras by adding further conditions, and in this spirit we in fact arrive at a description of these algebras. We also show, however, that every such algebra is a retract of some DF, F a free Stone algebra, so we conclude that there is no nice structural characterization of the projective Stone algebras along the line of Grätzer's programme.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 75 - 82
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

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