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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 C0 and D0 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 × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

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

1. Frink, O., Pseudo complements in semilattices, Duke Math. J. 29 (1962), 505–514.Google Scholar
2. G., Grâtzer, General lattice theory (Academic Press, New York, 1978).Google Scholar
3. P. V., Ramana Murty, Prime and implicative semilattices, Algebra Universalis 10 (1980), 31–35.Google Scholar
4. P. V., Ramana Murty and V. V., Rama Rao, Characterization of certain classes of pseudocomplemented semilattices, Algebra Universalis 4 (1974), 289–300.Google Scholar
5. Nemitz, W. C., Implicative semilattices, Trans. Amer. Math. Soc. 117 (1965), 128–142.Google Scholar
6. Varlet, J. C., Relative annihilators in semilattices, Bull. Austral. Math. Soc. 9 (1973), 169–185.Google Scholar
7. Varlet, J. C., On separation properties in semilattices, Semigroup Forum 10 (1975), 220–228.Google Scholar