Vol. 55, No. 1, 1974

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On (J, M, m)-extensions of Boolean algebras

Dwight Webster Read

Vol. 55 (1974), No. 1, 249–275

The class 𝒦 of all (J,M,m)-extensions of a Boolean algebra 𝒜 can be partially ordered and always contains a maximum and a minimal element, with respect to this partial ordering. However, it need not contain a smallest element. Should 𝒦 contain a smallest element, then 𝒦 has the structure of a complete lattice. Necessary and sufficient conditions under which 𝒦 does contain a smallest element are derived. A Boolean algebra 𝒜 is constructed for each cardinal m such that the class of all m-extensions of 𝒜 does not contain a smallest element. One implication of this construction is that if a Boolean algebra 𝒜 is the Boolean product of a least countably many Boolean algebras, each of which has more than one m-extension, then the class of all m-extensions of 𝒜 does not contain a smallest element. The construction also has as implication that neither the class of all (m,0). products nor the class of all (m,n)-products of an indexed set {𝒜t}tT of Boolean algebras need contain a smallest element.

Mathematical Subject Classification
Primary: 06A40
Received: 27 November 1972
Revised: 28 March 1973
Published: 1 November 1974
Dwight Webster Read