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Pseudocomplemented and Implicative Semilattices

Published online by Cambridge University Press:  20 November 2018

C. S. Hoo
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Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C 0 and D 0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D 0 f. In case a = 0, Murty and Rao [4] have shown that C 0 × D 0/f is isomorphic to L, and hence that C 0 × D 0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C 0 × D 0/f is isomorphic to L, and that C 0 × D 0/f is also implicative.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982, pp. 423 - 437
Copyright
Copyright © Canadian Mathematical Society 1982

References

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