Vol. 62, No. 1, 1976

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Ultrafilters and the basis property

Richard Alexander Sanerib, Jr.

Vol. 62 (1976), No. 1, 255–263

Three notions of a basis for an ultrafilter in a Boolean algebra are investigated in this paper, namely having an independent set of generators, a weakly independent set of generators and a weakly independent set of generators over a proper subfilter. In general these three notions are distinct, but for a Boolean algebra with an ordered base the latter two are equivalent. This paper shows that a large class of Boolean algebras do not possess ultrafilters with a basis, in particular no infinite homomorphic image of a σ-complete Boolean algebra has a nonprincipal ultrafilter with a basis. For Boolean algebras with an ordered base necessary and sufficient conditions on the order type of the base are given for the Boolean algebra to have the basis property.

Mathematical Subject Classification
Primary: 06A40, 06A40
Received: 26 June 1975
Revised: 3 November 1975
Published: 1 January 1976
Richard Alexander Sanerib, Jr.