Some Examples of Constructive and Non-constructive Extensions of the Countable Atomless Boolean Algebra

There are two main results concerning the nature of constructive extensions of the countable atomless Boolean algebra, say 28 a, which appear in [1]. These results are (I) if <& is a constructive extension of 23a, then each element of # is recursive open and regular, and (II) the characterization of the simple constructive extensions 28JJJ) as those for which U is recursive open and regular. We note that by extension 28 of 28 a we mean an extension of 28 a such that 28 is a subalgebra of the completion $a, the Boolean algebra of regular open sets [2]. The results (I) and (II) are true for constructive extensions of an arbitrary computable Boolean algebra; however, at that time the only known constructive extensions of 28 a were obtained by (II), while the only non-constructive extensions of 28 a were either uncountable or obtained from 2ft a by adjoining regular elements which were not recursive open, i.e., these extensions were non-constructive on the basis of cardinality or by (I). Eventually, in [3], the first author produced examples of non-simple constructive extensions of 28 a. In the present work we produce non-constructive extensions of 28 a each of whose elements is recursive open and regular. In addition, we produce for each recursive ordinal a a-chains with respect to (strongly) constructive extension over 28a. The main tool used in obtaining these results is the construction, for any constructive extension # of 28 a, of a strongly constructive extension of 6.