### Some notes concerning the homogeneity of Boolean algebras and Boolean spaces

by Geschke and Shelah. [GeSh:811]

Topology and its Applications, 2003

We consider homogeneity properties of Boolean algebras
that have nonprincipal ultrafilters which are countably
generated.
It is shown that a Boolean algebra B is homogeneous if it is
the union of countably generated nonprincipal ultrafilters and
has a dense subset D such that for every a in D the
relative algebra B restriction a:= {b in B:b <= a} is
isomorphic to B . In particular, the free product of
countably many copies of an atomic Boolean algebra is homogeneous.
Moreover, a Boolean algebra B is homogeneous if it satisfies
the following conditions:
(i) B has a countably generated ultrafilter,
(ii) B is not c.c.c., and
(iii) for every a in B setminus {0} there are finitely many
automorphisms h_1, ...,h_n of B such that
1=h_1(a) cup ... cup h_n(a) .

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