Intrinsically recursively enumerable subalgebras of a recursive Boolean algebra

Translated from Algebra i Logika, Vol. 31, No. 1, pp. 38–46, January–February, 1992.     

Special subalgebras of Boolean algebras

We consider eight special kinds of subalgebras of Boolean algebras. In Section 1 we describe the relationships between these subalgebra notions. In succeeding sections we consider how the subalgebra notions behave with respect to the most common cardinal functions on Boolean algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On subalgebras of Boolean interval algebras

We prove that the following three conditions are necessary and sufficient for a Boolean algebra to be embeddable into an interval algebra.

(i)

is generated by a subset such that for all .

(ii)

has a complemented subalgebra lattice, where complements can be chosen in a monotone way.

(iii)

is isomorphic to ClopX for a compact zero-dimensional topological semilattice such that for all .

     

On maximal subalgebras of the algebras of unary recursive functions

Under consideration are the algebras of unary functions with supports in countable primitively recursively closed classes and composition operation. Each algebra of this type is proved to have continuum many maximal subalgebras including the set of all unary functions of the class ε ² of the Grzegorczyk hierarchy.     

Classes of recursively enumerable sets and Q-reducibility

Translated from Matematicheskie Zametki, Vol. 45, No. 2, pp. 79–82, February, 1989.     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .