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Covers and complements in the subalgebra lattice of a Boolean algebra

Published online by Cambridge University Press:  17 April 2009

Ivo Düntsch
Ivo Düntsch
Affiliation:
Department of MathematicsUniversiti Brunei DarussalamNegaraBrunei Darussalam
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Abstract

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Section 1 addresses the problem of covers in Sub D, the lattice of subalgebras of a Boolean algebra; we describe those BA's in whose subalgebra lattice every element has a cover, and show that every small and separable subalgebra of P(ω) has 2ω covers in SubP(ω). Section 2 is concerned with complements and quasicomplements. As a general result it is shown that Sub D is relatively complemented if and only if D is a finite– cofinite BA. Turning to Sub P(ω), we show that no small and separable D ≤ P(ω) can be a quasicomplement. In the final section, generalisations of packed algebras are discussed, and some properties of these classes are exhibited.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 3 , December 1989 , pp. 371 - 379
Copyright
Copyright © Australian Mathematical Society 1989

References

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