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Effective presentability of Boolean algebras of CantorBendixson rank 1
 Journal of Symbolic Logic
, 1999
"... We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though F ..."
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We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite CantorBendixson rank.
Characterizing the Computable Structures: Boolean Algebras and Linear Orders By
, 2007
"... A countable structure (with finite signature) is computable if its universe can be identified with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study ..."
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A countable structure (with finite signature) is computable if its universe can be identified with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study the Boolean algebras of low Ketonen depth, both classically and effectively. Classically, I give an explicit characterization of the depth zero Boolean algebras; provide continuum many examples of depth one, rank ω Boolean algebras with range ω + 1; and provide continuum many examples of depth ω, rank one Boolean algebras. Effectively, I show for sets S ⊆ ω + 1 with greatest element, the depth zero Boolean algebras Bu(S) and Bv(S) are computable if and only if S \{ω} is Σ 0 n↦→2n+3 in the Feiner Σhierarchy. Making use of the existing notion of limitwise monotonic functions and the new notion of limit infimum functions, I characterize which shuffle sums of ordinals below ω + 1 have computable copies. Additionally, I show that the notions of limitwise monotonic functions relative to 0 ′ and limit infimum functions coincide.
DEPTH ZERO BOOLEAN ALGEBRAS
, 2010
"... Abstract. We study the class of depth zero Boolean algebras, both from a classical viewpoint and an effective viewpoint. In particular, we provide an algebraic characterization, constructing an explicit measure for each depth zero Boolean algebra and demonstrating there are no others, and an effecti ..."
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Abstract. We study the class of depth zero Boolean algebras, both from a classical viewpoint and an effective viewpoint. In particular, we provide an algebraic characterization, constructing an explicit measure for each depth zero Boolean algebra and demonstrating there are no others, and an effective characterization, providing a necessary and sufficient condition for a depth zero Boolean algebra of rank at most ω to have a computable presentation. 1.
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"... class is an e ectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of ..."
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class is an e ectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of