On the non-existence of free complete Boolean algebras

Hales, Alfred W.

doi:10.4064/fm-54-1-45-66

Cited by 55 publications

(31 citation statements)

References 5 publications

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“…This conjecture is, in effect, that no infinite free Boolean algebra has a free complete extension this has been verified by H. Gaifman in [3] and, independently and by different methods, by A. Hales, in [4].…”

Section: Free Complete Extensions Of Boolean Algebras George W Daymentioning

confidence: 96%

Free complete extensions of Boolean algebras

Day¹

1965

Pacific J. Math.

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Section: Free Complete Extensions Of Boolean Algebras George W Daymentioning

confidence: 96%

Free complete extensions of Boolean algebras

Day¹

1965

Pacific J. Math.

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“…This fact and Theorem 1 justify calling FD(\X\) the free completely distributive lattice on |A"| generators. It is interesting to note that Gaifman [2] and Hales [3] have shown that there does not exist a Boolean algebra over countably many generators. For finite sets X, the free distributive lattice on |A'| generators is often considered to be our FD(\X\) -[0, 2X) (see [1]).…”

Section: Free Completely Distributive Lattices George Markowsky1mentioning

confidence: 99%

Free Completely Distributive Lattices

Markowsky¹

1979

Proceedings of the American Mathematical Society

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Abstract. We show that the usual construction of the free distributive lattice on n generators generalizes to an arbitrary quantity of generators and actually yields a free completely distributive lattice. Furthermore, for an infinite number of generators the cardinality of the corresponding free completely distributive lattice is exactly that of the power set of the power set of the set of generators. . For a finite set, G, of n generators it is well-known that the free distributive lattice over « generators is isomorphic to the collection of all closed below subsets of the power set of G. It is also possible to find free distributive lattices over an infinite number of generators [7]. However, these are quite different objects.For a given set X of arbitrary cardinality, we denote by FD(\X\) the collection of all closed below subsets of the power set (2*) of X. We then have the following result.

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“…(It suffices to consider the Dedekind completion of the Boolean algebra from Theorem K; in [13] it is remarked that the method of constructing this Boolean algebra is due to Hales [5].) 2.2.…”

Section: Generators and Complete Generatorsmentioning

confidence: 99%

“…In [5] it was proved that if α is an infinite cardinal, then there exists no free complete Boolean algebra with α free complete generators. A similar result was proved in [10] for complete lattice ordered groups and in [11] for complete vector lattices.…”

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confidence: 99%