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Imbedding Infinitely Distributive Lattices Completely IsomorphicallyInto Boolean Algebras

Published online by Cambridge University Press:  22 January 2016

Nenosuke Funayama
Nenosuke Funayama*
Affiliation:
Yamagata University
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It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 15 , August 1959 , pp. 71 - 81
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

References

[1] Birkhoff, G., Lattice Theory, Revised edition, Amer. Math. Soc. Colloq. publ., vol. 25 (1948).Google Scholar
[2] Birkhoff, G., On rings of sets, Duke Math. J., vol. 3 (1937), pp. 443–454.CrossRefGoogle Scholar
[3] Funayama, N., Imbedding partly ordered sets into infinitely distributive complete lattices, Tohoku Math. J., vol. 8 (1956), pp. 54–62.CrossRefGoogle Scholar
[4] Peremans, W., Embedding of a distributive lattice into a Boolean algebra, Nederl. Akad. Wetensch. Proc. Ser. A, 60 = Indag. Math., vol. 19 (1957), pp. 73–81.CrossRefGoogle Scholar
[5] Schmidt, J., Die transfiniten Operationen der Ordnungstheorie, Math. Ann., vol. 133 (1957), pp. 439–449.CrossRefGoogle Scholar
[6] Smith, C. E. and Tarski, A., Higher degree of distributivity and completeness in Boolean algebras, Trans. Amer. Math. Soc, vol. 84 (1957), pp. 230–257.Google Scholar
[7] Smith, C. E., A distributivity condition for Boolean algebras, Ann. Math., vol. 64 (1956), pp. 551–562.Google Scholar