This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995

Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.

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 Cantor-Bendixson 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 Cantor-Bendixson rank.

The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are a number of topics that I will just mention in order to give some indication of what lies beyond what is discussed here. 1. The Ziegler spectrum 2. Various dimensions 3. These dimensions for artin algebras 4. These dimensions in general 5. Duality 6. The complexity of morphisms in mod-R 7. The Gabriel-Zariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pure-injectives Suppose that M is a submodule of N . Consider a finite system \Sigma n i=1 x i r ij = a j (j = 1; :::m) of R-linear equations over M : that is, the r ij are in R, the 1 a j are in M and the x i are indeterminates. Suppose that there is a solution b 1 ; ...

An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is oe-complete and satisfies the following property: Whenever L is isomorphic to an interval, [a; b], in L then L is isomorphic to each interval [c; d] with c a and d b (resp. the same condition as above only under the assumption that all elements a, b, c, d are central in L). Let us denote by Inthom (resp. Inthomc) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous orthomodular lattices). We first show that the class Inthom is considerably large --- it contains any Boolean oe-algebra, any block-finite oe-complete orthomodular lattice, any Hilbert space projection lattice and several other examples. Then we prove that L belongs to Inthom exactly when the Cantor-Bernstein-Tarski theorem holds in L. This makes it desirable to know whether there exist oe-complete orthomodular lattices which do not belong to Inthom. Such ex...

A countable structure (with finite signature) is computable if its universe can be identi-fied 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 Σ-hierarchy. 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.

isomorphic. We further generalize this result to oe-complete MV-algebras under the condition that the bounds a; b are boolean elements. We compare this result to the theorem proved by Jakub'ik in the setting of complete MV-algebras [4]. We also prove another generalization of Cantor-Bernstein theorem for oe-com- plete orthomodular lattices. We relate this result to the so-called "Tarski cube phenomenon": There is a Boolean algebra A such that A 2 is not Boolean isomorphic to A but A is Boolean isomorphic to A 3 (see [3] and [5]). This Boolean algebra obviously cannot be oe-complete. As a consequence of the generalized Cantor-Bernstein theorem, we show that the Tarski cube phenomenon cannot occur also for oe-complete orthomodular lattices. References

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.

Abstract. We prove that, for every cardinal number α ≥ c, there exists a metrizable space X with |X | = α such that for every pair of quasiorders ≤1, ≤2 on a set Q with |Q | ≤ α satisfying the implication q ≤1 q ′ = ⇒ q ≤2 q ′ there exists a system {X(q) : q ∈ Q} of non-homeomorphic clopen subsets of X with the following properties: • q ≤1 q ′ if and only if X(q) is homeomorphic to a clopen subset of X(q ′), • q ≤2 q ′ implies that X(q) is homeomorphic to a closed subset of X(q ′ ) and • ¬(q ≤2 q ′ ) implies that there is no one-to-one continuous map of X(q) into X(q ′).