Computable Boolean algebras

Abstract

Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T 2 ]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T 3 ], showed that every low 2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low 3 , or even low 4 , has a computable copy. The results of [D-J] and [T 3 ] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case). Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmel's isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low 2 Boolean algebra.