R-maximal Boolean algebras

Metakides and Nerode in [2] suggested the study of what they termed the lattice of recursively enumerable substructures of a recursively presented model. For example, Metakides and Nerode in [3] introduced the lattice of of recusively enumerable subspaces, , of a recursively presented vector space V ∞ . The similarities and differences between and ℰ, the lattice of recursively enumerable subsets of the natural numbers N as defined in [9], have been studied by Metakides and Nerode, Kalantari, Remmel, Retzlaff, and Shore. In [6], we studied some similarities and differences between ℰ and the lattice of recursively enumerable sub-algebras of a weakly recursively presented Boolean algebra and this paper continues that study. A weakly recursively presented Boolean algebra (W.R.P.B.A.) , , consists of a recursive subset of N , ∣ ∣, called the field of , and operations (meet), (join), and (complement) which are partial recursive and under which becomes a Boolean algebra. We shall write and for the zero and unit of . If S is a subset of , we let ( S )* denote the subalgebra generated by S . Given sub-algebras B and C of , we let B + C denote ( B ⋃ C )*. A subalgebra B of is recursively enumerable (recursive) if { x ∈ ∣ ∣ x ∈ B } is a recursively enumerable (recursive) subset of ∣ ∣. The set of all recursively enumerable subalgebras of , , forms a lattice under the operations of intersection and sum (+).