Some notes concerning the homogeneity of Boolean algebras and Boolean spaces

AbstractIn this article 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∈D the relative algebra B↾a:={b∈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∈B⧹{0} there are finitely many automorphisms h1,…,hn of B such that 1=h1(a)∪⋯∪hn(a).These results generalize theorems due to Motorov [Russian Math. Surveys 44 (16) (1989) 190–191] on the homogeneity of first countable Boolean spaces.Finally, we provide three constructions of first countable homogeneous Boolean spaces that are linearly ordered. The first construction gives separable spaces of any prescribed weight in the interval [ℵ0,2ℵ0]. The second construction gives spaces of any prescribed weight in the interval [ℵ1,2ℵ0] that are not c.c.c. The third construction gives a space of weight ℵ1 which is not c.c.c. and which is not a continuous image of any of the previously described examples