Computability of boolean algebras and their extensions

Abstract

A Boolean algebra is computable if there is a one-to-one enumeration ( O n ) nϵN of its domain which associates recursive functions with sup, inf and complement. A computable Boolean algebra with B enumeration ( U n ) is a constructive extension of its computable sub-algebra algebra U with enumeration ( O n ) if there is a recursive function h such that O n = U h( n) . Let U be a computable Boolean algebra with enumeration ( O n ) whose elements are the clopen sets in some Boolean space T . A subset U of T is recursive open (respectively, recursively regular open) iff there are recursive functions f and g such that U =∪ nϵ N O f(n) and U −1 = ∪ nϵ N O g(n) ( respectively, U = (∪ nϵ N O f(n)) −1−1 = (∩ nϵ N O g(n)) −1−1) . Since the regular open sets form a minimal completion of U (see the introduction), the concepts represent two attempts to define “the recursive elements of” this completion. Detailed motivation for the definitions is included in the intorduction. The recursively regular open sets form a Boolean algebra (under the operatons which make the regular open sets into a Boolean algebra). A simple extension of U obtained by adjoining a regular open set U can be given the structure of a constructive extension of U iff recursive open. The class of recursive open sets which are regular is the union of all (simple) constructive extensions of U which consist of regular open sets. Henceforth, assume T is the Cantor space, so that U is atomless. The recursive open sets which are regular do not form a Boolean algebra. U possesses computable extension which are constructive and others which are not constructive. The Boolean algebra generated by the recursive open sets which are regular the algebra of recursively regular open sets, so that the two attempts to define “the recursive elements of” the completion coincide.