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ON THE LATTICE OF II10 CLASSES By FARZAN RIAZATI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001 Copyright 2001 by Farzan Riazati ACKNOWLEDGMENTS I wish to express my gratitude to my advisor, Professor Douglas Cenzer. I also wish to thank Andre Nies. TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................. iii LIST OF TABLES ................................. v ABSTRACT ... ... .. .. ... ... .. ... . . .. . .. ... vi CHAPTERS 1 PRELIMINARIES ........................... 1 1.1 Terminology and Notation ...................... 1 1.2 Existence and Examples ........................ 5 1.3 Basic Properties ......................... 8 1.4 Appearances and Applications .................... 11 1.5 Stone Representation and Computable Boolean Algebras . 15 2 LOCAL PROPERTIES OF H? CLASSES . . . . . .... .. 17 2.1 Post Program and the Lattice Cn . . . . . . .. .. 17 2.2 Principal Ideals of H? Classes . . . . . . . ... .. 17 2.3 The World of Simple IIH Classes . . . . . . ... .. 21 2.3.1 Minimal Extensions of H? Classes . . . . ... ..22 2.3.2 Quasiminimal IIH Classes . . . . . . ... .. 30 2.3.3 Fully Complemented Principal Ideals . . . ... ..33 2.3.4 Where There Are No Complements; Nerode's Theorem 38 2.4 Principal Covering Filters . . . . . . . . ... .. 41 2.5 The Splitting Property . . . . . . . . . ... .. 44 3 GLOBAL PROPERTIES OF II? CLASSES . . . . . .... .. 48 3.1 Definability, and Automorphisms . . . . . . .... .. 48 3.2 Homogeneity and Embeddings . . . . . . . .... .. 51 REFERENCES . . . . . . . . . . . . . . . . ... .. 58 BIOGRAPHICAL SKETCH . . . . . . . . . . . . .... 62 LIST OF TABLES Table page 2.1 Initial principal ideals of ^ ....... .......................18 2.2 Principal filters of * . . . . . . . . . . . . .. .. 22 2.3 Structure of principal ideals of C .. ................... ..32 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE LATTICE OF IIo CLASSES By Farzan Riazati August 2001 Chairman: Professor Douglas Cenzer Major Department: Mathematics Effectively closed sets, as modeled by H' classes, have played an important role in computability theory going back to the Kleene basis theorem [1955]. Many of the fundamental results about 110 classes and their members were established by Jockusch and Soare in [1972]. Cenzer et. al [1999] is a short course on IIH classes. The n? classes occur naturally in the application of computability to many areas of mathematics. Cenzer and Remmel [1998] is a recent survey with many examples. Minimal and thin IH? classes were investigated by Cenzer et al. [1993]. This dissertation is a comparative study of the lattice Cn of H? classes with the lattice of computably enumerable (c.e.) sets. The work in this thesis concerns the lattice of H? classes (modulo finite difference), compared and contrasted with the lattice of c.e. sets. The notion of a minimal extension Q of a class P is defined to mean that there is no class strictly between P and Q. Previously only trivial examples were known, but here we give general conditions under which P has a minimal extension. Recently initial segments of the lattice (that is, subsets of a given set) have been studied. It was shown, in contrast to the lattice of c.e. sets, that a finite lattice can be realized which is not a Boolean algebra; in particular, any finite ordinal can be realized. This thesis announces an improvement of these results by constructing a 11 class P such that the family of subclasses of P is isomorphic to the smallest infinite ordinal (w). Also studied are definability of various properties (such as finiteness) and invariance under automorphism. CHAPTER 1 PRELIMINARIES 1.1 Terminology and Notation We begin with some basic definitions. Let w = {0, 1,2, ...} denote the set of natural numbers. For any set E, E<' denotes the set of finite strings (a(0),... ,c(n 1)) of elements from E. And Ew denotes the set of countably infinite sequences from E. For a string cr = (u(0), (1),..., u(n 1)), luall denotes the length n of a. The empty string has length 0 and is denoted by 0. A string of n many k's is denoted by kV. For m < i la, a m is the string (a(0),..., a(m 1)). We say cr is an initial segment of r (written ar < r) if a = rfm for some m. Given two strings a and r, the concatenation of a and r, denoted by crUr (or sometimes just ur), is defined by orT = (a(O),U(1),...,a(m 1),r(O),r(1),...,r(n 1)), where Ilall = m and IT11 = n. We write aa for a"(a) and aau for (a)'. For any x E EW and any finite n, the initial segment x[n of x is (x(0),...,x(n 1)). For a string a E x E ES, we have aux = (o(O),..., a(n 1), x(O), x(1),...). Given strings ar and r of length n, we let a r = (ar(O),r(O),..., a(n 1), r(n 1)); if lull = n + 1 and 1 rll = n, then a r = (urn r)a(n). Given two elements x,y of E, x y = z where z(2m) = x(m) and z(2m + 1) = y(m). We need to code a string a E w 2 For each a E wk, we let < a >= [< a[(k 1) >,a(k)], where <>= 0, and < m >= mn for each mn E w. A tree T over Ei< is a set of finite strings from E< which contains the empty string 0 and which is closed under initial segments. We say that r E T is an immediate successor of a string a E T if = a'a for some a E E. Since our alphabets will always be countable and effective, we may assume that T C w<^. Such a tree is said to be wbranching since each node has potentially a countably infinite number of immediate successors. We shall identify T with the set {< a >: oa E T}. Thus we say that T is computable, computably enumerable, etc. if {< a >: a E T} is computable, computably enumerable, etc. For a given function g : w T is said to be finite branching if T is gbounded for some g that is, if each node of T has finitely many immediate successors. Observe that this is equivalent to the existence of a bounding function h such that a(i) < h(i) for all a E T and all i < Ial. T is said to be computably bounded (c.b.) if it is gbounded for some computable function g. As above, this is equivalent to the existence of a computable bounding function h such that acr(i) < h(i) for all a E T and all i < ljaoI. If T is computable, then this is also equivalent to the existence of a partial computable function f such that, for any a E T, a has at most f(a) immediate successors in T. A computable tree T is said to be highly computable if it is also computable bounded. For any tree T, an infinite path through T is a sequence (x(O), x(l),...) such that x \n E T for all n. We let [T] denote the set of infinite paths through T. It is important to note here that we consider a 110 set to signify a subset of w and in general a 110, E or A set is a subset of w with the appropriate form of definability in the arithmetical hierarchy [23]. We denote by card(A) the cardinality of the set or class A. Given two trees S and T contained inw the amalgamation of Pand Q, P Q, by P Q = { y :x C PAy Q}. Then it is clear that P 0 Q =[S T. More generally, define the infinite amalgamation OiSi to be those strings a such that for each i, (a([i,O]),a([i, 1]),.. .,a([i,j])) E S,, where j is the maximum such that [i,j] < ]Iall. Then [0iS,] is isomorphic to the direct product rili[Si]. We also wish to consider the following notion of disjoint union. Given two trees S and T contained in w I0 classes P = [S] and Q = [T] P E Q = {Ox x P U {ly : y Q}. It is easy to see that [S e T] = [S] [T]. Clearly S T is bounded if and only if both S and T are bounded and similarly for the other notions of boundedness. More generally, the infinite disjoint union DiQi may be defined to be {(i) y y y Qi} for unbounded classes Qi. A node a of the tree T C w<` is said to be extendible if there is some x C [T] such that a < x. The set of extendible nodes of T is denoted by Ext(T). Ext(T) may be viewed as the minimal tree S such that [5S] = [T]. A node a E T is said to be a dead end if a ( Ext(T), that is, if a has no infinite extension in [T]. We are interested in 110 classes in the spaces {0, 1}'(the Cantor space) and wa (the Baire space). The topology on wu is determined by a basis of intervals 1(a) = {x C w : cr < x}. Notice that each interval is also a closed set and is therefore said to be a clopen set. Moreover the clopen subsets of the Cantor space are just the finite unions of intervals. The CantorBendixson derivative D(P) of a compact subset P of {0, 1}" is the set of nonisolated points of P. Thus a point x E P is not in D(P) if and only if there is some open set U containing x which contains no other point of P. Equivalently, x D(P) if and only if there is some closed set U such that U fl P = {x}. Another useful observation is that, for any compact set P, D(P) is empty if and only if P is finite. The iterated CantorBendixson derivative D'(P) of a closed set P C X is defined for all ordinals a by the following transfinite induction. D(P) = P; D+l'(P) = D(D`(P)) for any a; DA(P)= no<\ Da(P) for any limit ordinal A. The CantorBendixson rank of a countable closed set P is the least ordinal a such that D'+'(P) = 0. The (effective) CantorBendixson rank of a point x E X is the least ordinal a such that, for some H class P, Dc(P) = {x}. For a E {0, 1}<', the interval I(a) = {x E 2 : or < x}. We denote by )C(2w) the collection of all clopen subsets of 2W i.e. the finite unions of intervals, I(rai) U ... U I(on). We also note that for a closed subset Q of 2w, D(Q) = {x: (VU E KA(2w))(x E U 4 (3y 5 x)(y E U n Q)}. We refer the reader to Odifreddi [39], or Soare [47] for the basic definitions of computability theory. In particular let O, be the partial computable functional with index i and let Oi,s be the computation of 4j for s steps, so that the function @,, is uniformly primitive computable. We write 0,(o,) . if (3s)(4e,s(r) ) and ,e(ca) t if not 0, (a) 4.. The computably enumerable sets are enumerated as We = {n: ,e(n) i.}. The computable functions of type two, which take both number and function variables can also be enumerated, as 4). Here we write o(n) to denote the result of computing e on a number variable n and a function variable x. The result of computing V(n) for s steps is written O,8(n) and uses only the first s values of x. Given two sets A and B, we write A reducible to B is there is oneone computable function f such that x E A iff f(x) is in B. We write A =m if A identify W with it's characteristic function. Let 0 be the Turing degree of the empty set. 1.2 Existence and Examples The existence of non trivial [ classes is a consequence of the following classic result. Theorem 1.2.1 (Konig [30]) An infinite tree in which every node has only finitely many immediate successors has an infinite branch. Consider the proof of the above lemma. Let T be such a tree. We define an infinite branch by induction. We start with uo. Given an with infinitely many extensions on T, let an,+ be an immediate successor of on with infinitely many extensions on T. It exists because a, has infinitely many extensions on T, but only finitely many immediate successors. Thus at least one of them must have infinitely many extensions on T. An analysis of the proof of the Konig's Lemma provides that : Corollary 1.2.2 (Kreisel's Lemma) An infinite computable binary tree has a A infinite branch. This result has been improved by Shoenfield [44] who proved that there is always a branch of Turing degree less than 0'. However the next result is much stronger. Theorem 1.2.3 (Jockusch, Soare [25]) The Turing degrees a such that a' = 0', (low degrees), form a basis for H' classes. Every H class has an element of low degree. Recall that [T] denotes the set of infinite paths through a tree T. That is, [T] = {x E 2w : (Vn)(x[n T)}. For any tree T, the set of extendible nodes of T is defined by Ext(T) = {a: (3x E [T])(a < x)}. For a II? class P, which is the set of infinite paths through some computable tree T, we denote the subtree Ext(T) of T by Tp. The II? class P is said to be decidable if Tp is computable, or equivalently, if P = [T] for some computable tree T with no dead ends. The task of showing that a particular class P is H' is often simpler than producing a computable tree T with P = [T]. It suffices to have an effective procedure which, given an oracle for a function x P, discovers that x P within finitely many steps, and runs forever on any oracle x C P. Definition 1.2.4 An element x of a H' class P is said to be isolated if there is some a such that P fn I(a) = {x}. As we shall see, isolated paths are often computationally trivial. In fact, the computational complexity of the CantorBendixson derivative of a IIH class is an important feature. It is often useful to view II classes in 2<" via their complements. There are many trees representing a particular class P, but there is one tightest representation via the strings not in P, which brings us to the following definition. Definition 1.2.5 A subset G of 2< is called a 2 Furthermore, G C 2<' is called a 2<' c.e. ideal if G is a c.e. set and is a 2<' ideal. We have the following correspondence between IIH classes and 2 Lemma 1.2.6 Let P C 2w, Then P Cn  2<"w \ Tp is a 2<" c.e. ideal. Proof: Suppose P = [T] is a n? class where T is a computable tree. 2 \ Tp is a c.e. set, since, by Konig's lemma, a E 2w \Tp (3n)(Vr E {0, 1}")aTT T By definition of Tp, if r E Tp then so are all of its initial segments. Therefore a E 2<' \ Tp implies that all of a's extensions are too. Finally if a E Tp the some extension of a must be there too. (Indeed a has extensions of all lengths). In particular, one of a'0 or al must be in Tp by the downward closure property. Therefore 2 II? classes may also be characterized in terms of c.e. ideals and filters in the countable Boolean algebra C(2) of clopen subsets of 2". Let Z(P) = {U E KC(2) : U n P= 0}, and Up = {u E KC(2w) : P C U}. Observe that U e I(P) U U UpK, and Z(P) is clearly an ideal, and Up is a filter, for each P C 2w. Lemma 1.2.7 The following are equivalent: i) P is II class. ii) Z(P) is a c.e. ideal. iii) Up is a c.e. filter. Proof: i) => ii) Let G = 2w \ Tp, since P is a H? class, G is a c.e. ideal. Let U = I(Cri)UI(72)U.. .UI(an), then U E Z(P) 4= (Vi < n)o E G. ii) = iii) This follows from U E Z(P) = U E p. iii) => i) Forany a, wehave a E G = I(acr) E Z(P), so G is a c.e. ideal, which makes P a H1 by the previous lemma. There are IIH? classes with Tp of every c.e. degree. Given any c.e. set, we let 0'1 in the subtree of the extendible nodes, by branching off the limit path 0' when and only when n E. Let E be any c.e. set, define a 11? class by setting Tp to contain 0' for every n, and for every m satisfy; n Em ==> 0"1m E Tp. Thus [Tp] = {0fO} U {0'1' : n E}. This will be an important example for later developments, when the above c.e. set E is a maximal set. The fundamental example of H? classes is the separating class of two separable c.e. sets. The sets separating two disjoint c.e. sets A and B form a [I? class: C E SA,B 4 (Vx)[(x E A 4 x E C)&(x E B + x C)]. Explicitly, a computable tree whose branches are exactly the members of SA,B is the following: x E TA,B 4==* x is a sequence correct up to Iixll where 'x is correct up to stage s' means that for every i < ]xa, if i E As then x(i) = 1, and if i E B, then x(i) = 0. In other words, we seal off a branch of TA,B as soon as we discover it is incorrect. Note that TA,B has an infinite branch if and only if A and B are disjoint. Moreover an infinite branch of TA,B is the characteristic function of a set separating A and B. Simpson poset is used to give a universal example of H' classes. Definition 1.2.8 Simpson Poset of degrees Simpson order << is defined in [45] as ; b << a every infinite tree T C 2 particular, 0 << a means that every nonempty II class of sets has an element of degree < a. There is a universal 11 class U C 2w defined as; U = {x E 2w: (Ve)[x(e) (e)]}. This class U is a class of sets which separate {e: 0,(e) = 0} and {e: 0,(e) = 1}. Obviously U J 0 has no computable member and is quite big. It also has the property that the degree of elements of U coincide with the degree of complete extensions of Peano arithmetic. It has also been shown that, for any degree a, 0 << a if and only if there exists f U of degree < a. Thus the degrees of elements of U are exactly the degrees >> 0, which means that each element of U can compute an element of any nonempty H' class, hence the universality of U. 1.3 Basic Properties When a II? class has no computable element, the elements are generally of lower computational complexity. Jockusch and Soare [25] shown that a IIn class without computable members has cardinality 210. Since every isolated infinite branch of a computable tree is computable, if there are no infinite computable branches every branch splits, and the number of infinite branches is 21. Jockusch and Soare [25] also proved that A IIH class without computable members is meager. It follows that singleton II? classes are topologic statements about their unique members. Lemma 1.3.1 For any x E 2, the following are equivalent: i) x is computable. ii) {x} is a [II class. iii) The CantorBendixson rank of x is zero. Basis Theorems are the computability theorist's Choice principles. These the orems are often informally stated as: "Every simply definable class has a simply definable member" Kleene observed that, given the set of extendible nodes Ext(T) as an oracle, one can compute an infinite path through T (if one exists) by letting; x(n) be the least k such that (x(O),..., x(n 1)) E Ext(T). For an arbitrary com putable tree, Ext(T) is E. The type of "simply definable element" depends on the complexity of the "simply definable class". Theorem 1.3.2 (Kleene, KreiselShoenfield, CenzerRemmel) For each H? class P C W, i) P has a member computable from some E' set; ii) If P is bounded, then P has a member computable from 0"; iii) If P is computably bounded, then P has a member computable from 0'; iv) If P is decidable, then P has a computable member; v) If P is highly polynomialtime decidable, then P has a ptime member. Forcing with IH classes, is another witness to the richness of the structure of subclasses of a H class under inclusion. This suggests similar questions to Post's program for the c.e. sets. In case of computably bounded HI classes, more can be said about the "simply defined element": Theorem 1.3.3 (Jockusch and Soare[24]) Each nonempty c.b. 11H class P contains: i) an element of low degree. ii) an element of c.e. degree. iii) two elements whose degrees have infimum zero. iv) an element of hyperimmunefree degree. Part (i) above which is known as the, Low Basis Theorem, is proved (like parts (iii) and (iv)) by forcing with nonempty H' subclasses of P. This is also interesting for methodological reasons. It is the first time the "information content" of a ll class P, is used to achieve results on its "computational content". (Once again more information about the II class P, would provide finer results). Theorem 1.3.4 Kleene Basis Theorem Let P be a countable 11H class of functions. Then i) P has a hyperarithmetic member. ii) If P is bounded, then P has a member computable from 0'. iii) If P is computably bounded, then P has a computable member. The following result may not look like a Basis Theorem, but it shows that nonempty computably bounded 11 classes are really "computably bounded". This result is used in second order arithmetic. For any sequence Co, C1i,... of noncomputable sets and any nonempty c.b. 11 class P, has a member x such that no C; is Turing reducible to x. The last and the most exciting Basis theorem is from Cenzer, Downey, Jockusch and Shore[7] which relates the computational complexity of an element x of a II? class P, to its topological standpoint in P. Definition 1.3.5 A Hl? class P is thin, if every subclass Q C P is relatively clopen in P, that is there exists a clopen set U such that Q = P n U. Definition 1.3.6 A 117 class M, is called a minimal H' class if for each H' subclass P C M, either M \ P is finite or P is finite. Lemma 1.3.7 (Cenzer et al. [2]) For any thin H' class P and any element x of P, x is isolated in P if and only if x is computable. Lemma 1.3.8 (Cenzer et al. [7]) Let P be a H1 class. If P is thin, and D(P) is a singleton, then P is minimal. If P is minimal and infinite, then D(P) is a singleton. Theorem 1.3.9 (Cenzer et al. [2]) Existence of Thin Classes. For every recursive ordinal a, there is a thin Hl' class Pa with CantorBendixson rank a, moreover P, may be taken to be the set of paths through a computable tree with no dead ends. Variuos construction can be carried out under general conditions specified in the next result. Theorem 1.3.10 (Cenzer, Downey, Jockusch and Soare[7]) Let P be a H' class of sets and suppose x E P. If x has rank 1 in P, then x 1.4 Appearances and Applications Logical theories and their extensions are the first applications. Since the class of sets separating two disjoint c.e. sets is a nonempty H1 class, so are: i) the set of consistent extensions of a given consistent theory. ii) the set of complete extensions of a given consistent theory. There are converses known to these results: Theorem 1.4.1 (Hanf [22]) The class of degrees of members of a given II? class coincides with the class of degrees of complete extensions of some finitely axiomatizable firstorder theory. Theorem 1.4.2 (Scott [43) If T is a consistent extension of PA, the sets computable in F form a basis for IHl classes. Theorem 1.4.3 Solovay [49] The following are equivalent: i) a is the degree of a consistent extension of PA ii) a is the degree of a complete extension of PA iii) D(5 a) is a basis for II classes. The set P(T) of computable extensions of a given axiomatizable theory F in first order logic is always a IIH class. For the theory F = PA (Peano Arithmetic), this provides the historically first example of a nonempty IIH class with no computable member. It can be shown that PA does not have maximal consistent extensions of low degree, c.e. degree, hyperimmunefree degree, etc. Theorem 1.4.4 (Ehrenfeucht [19]) Each II? class P C 2' can be represented in the form P(F) for some axiomatizable propositional theory F. For a decidable theory F, P(r) is a decidable IIH class and every decidable II? class of sets can be represented by the set of complete extensions of a decidable propositional theory. It follows that every decidable consistent theory has a decidable complete extension. A minimal decidable H class P, corresponds to a decidable theory T with exactly one undecidable complete extension and such that any axiomatizable extension of T is a principal extension of theory T. The complete extensions of an axiomatizable theory can be viewed as the maximal ideals of the c.e. Lindenbaum Boolean Algebra. Thus every decidablee) H1' class of sets may be represented as the family of maximal ideals of a c.e. computablee) Boolean Algebra. The next set of examples including challenging open questions are in Commutative rings. The collection of all prime ideals of a c.e. commutative ring with unity composes a c.b. H class. The example of Boolean rings shows that any c.b. H class of sets may be represented as the set of prime ideals of some computable ring. Theorem 1.4.5 (Friedman, Simpson and Smith [20]) Each II' class P of separating sets may be represented as the set of prime ideals of a computable commutative ring with unity. Problem 1.4.6 Does every c.b. H1 class represent the set of prime ideals of some computable commutative ring with identity? Graph theory, is another major application of 11 classes. The best source on problems in computable graph theory, is the Handbook of Recursive Mathematics [12]. Some properties of thin and minimal NO classes, have interesting representations in terms of graph coloring. If a thin HO class P is the set of kcolorings of some computably enumerable graph G, then for any computable coloring f of a computable subgraph H, the set of extensions of f to G is a H1' subclass of P and thus there is a finite subgraph G1 and a coloring g, of Gi such that the extensions of f are exactly the extensions of gi. For a minimal class P and a computable coloring f of a computable subgraph H, we see that either there are only finitely many kcolorings which extend f or all but finitely many kcolorings of G extend f. Partially ordered sets of finite width and their decomopositions, is another solution set representation problem. Theorem 1.4.7 (Dilworth [9]) Every finite poset of width k can be decomposed as the union of k chains. Where the width of a finite poset is the cardinality of the largest antichain. There is a natural dual to this theorem which says that every poset of height k can be covered by k antichains. The family of all decompositions of a given computable poset as the union of k chains (or anti chains) (k fixed) can be represented as a c.b. II' class. For the special case when P is the class of separating sets of a disjoint pair of c.e. sets, this has been answered by Cenzer and Remmel in [12], and independently by Hirst. The solution sets to many problems on the computable continuous functions on polish spaces, are c.b. II classes. The connection here is the following observation: The graph of a computably continuous function on a Polish space is always a 'II class. And for some Polish spaces, computably continuous functions have decidable II1 graphs, and each function with a II? graph is computably continuous. For each computably continuous function f, the following sets are II classes. i) The set of zeros of f. ii) The set of points where f attains a max or min. iii) The set of fixed points of f. iv) the complement of the basin of attraction of a computable periodic point of f, i.e. the Julia set of a computably continuous function f. The first two problems can represent any II? class in the given space. The fixed point problem can represent any II? class except for the space [0, 1] where the class must have a computable member. And the last problem can represent any class which is bounded and has a computable max and min element. Ramsey theory, index sets, are among the other applications. The recent paper of Cenzer and Jockusch [9], has a detailed analysis of the current state of research on fl? classes, including Ramsey theory. 1.5 Stone Representation and Computable Boolean Algebras Theorem 1.5.1 Every ideal on a Boolean algebra can be extended to a prime ideal. The Prime Ideal Theorem, is a weaker version of the axiom of choice, which is often used in many proofs in algebra and topology in place of full AC (compacti fication theorems, HahnBanach theorem,...). It also gives the following important representation theorem for Boolean algebras. Theorem 1.5.2 Every Boolean algebra is isomorphic to a field of sets. Let B be a Boolean algebra. We let S = {p : p is an ultrafliter on B}. For every u E B, let Xu be the set of all p E S such that u E p. Let T = {Xu : u E B} Let us consider the mapping 7r that 7r(u) = Xu. Clearly 7r(1) = S and 7r(0) = 0. It follows from the definition of ultrafilter that 7r(u.v) = 7r(u) n 7r(v), 7r(u + v) = 7r(u) U 7r(v), 7r(u) = S \ 7r(u). thus 7r is a homomorphism of B onto .T(and T is a field of sets). It remains to show that 7r is one to one. If u v then using the prime ideal theorem, one can find an ultrafilter p on B containing one of these two elements but not the other. Thus 7r is an isomorphism. Definition 1.5.3 The Stone space of a Boolean algebra B is the above space S with the topology given by the base F in the above theorem. With the above terminology, the Stone space is a compact Hausdorff space with a base of clopen sets. The Boolean algebra B is isomorphic to the algebra of all clopen sets of its Stone space. If the Boolean algebra B is countable then B is isomorphic to the Boolean algebra RC(P) of relatively clopen sets of a closed class P C 2. 16 Downey et al. [7] have proved an effective versions of these results and used it to transfer results on H[ classes to results on Boolean algebras which can be obtained as the quotient of a computable Boolean algebra by a c.e. equivalence relation. Among others they have discovered the interpretations of thinness and CantorBendixson derivative in computable Boolean algebras. CHAPTER 2 LOCAL PROPERTIES OF NO CLASSES 2.1 Post Program and the Lattice n It was first observed in Myhill [34] that the collection of all c.e. sets { We}e16 forms a lattice under inclusion, =< {We}e'; C>. The collection of all H classes of 2W forms a lattice under inclusion C relation and is denoted by n =< {Pe} .E ; C>. For a recent work on the Medvedev reductions see Cenzer and Hinman [8]. In this chapter we focus on the lattice Cn. Post [38] initiated studying the relationship between the Turing degree of a c.e. set, and its structure as a set with respect to set inclusion. For example Post defined a c.e. set A to be simple if its complement is infinite, but contains no infinite c.e. set. Post's Program was to find connections between the set theoretic structure C of a c.e. set and its Turing degree. We consider analogous questions in the lattice Cn, that resemble the Post's Program. We will compare and contrast with the analogous notions for the c.e. sets. 2.2 Principal Ideals of H Classes The IIH classes form a distributive lattice with smallest and greatest element, such that the clopen sets are the only complemented elements. As in the case of , these properties are far from characterizing the structures n and Cf. We will study the structure of the initial segments, and also we look at what structures may be realized as substructures of the lattice of H classes. Definition 2.2.1 n(P) is the principal ideal of Cn generated by the H' class P, i.e. Cn(P) = {R E nC : R C P}. We also denote by C(P) the lattice of II subclasses of P modulo finite differences. In fact C(P) is the quotient lattice of n(P)/1, where Z is the ideal of finite H' classes. Definition 2.2.2 [P, Q] is the interval of n between the H' classes P and Q, i.e. [P,Q] = {R En: P C R C Q}. Similarly, we use [P,Q]*, for the quotient [P,Q]/I, for the interval of P and Q in Ln. An ideal or filter itself forms a lattice under the induced operations. The structure of principal ideals C (P) can be highly varied. However the following important cases are already known. For a minimal H? class M, n(M) is isomorphic to the collection of finite, cofinite subsets of w, and C*(M) 0 {,1}. On the other hand n(P) for a thin 11' class P is lattice isomorphic to a subalgebra of the Boolean algebra of clopen subsets of 2', consisting of those clopen sets that have nonempty intersection with the closed set P, it follows that C(P) is also a Boolean algebra. The situation is summarized in the following table: Table 2.1: Initial principal ideals of * H? Class P The Principal Ideal Generated by P P minimal C(P) 0 {0,1} P thin n(P) Boolean Algebra The following idea is central in characterization of finite lattices that can be re alized as initial segments in Cn. In fact the next two theorems classify some of the principal ideals of the lattice n. Definition 2.2.3 The lattice (, <) satisfies the dual reduction property if for any a,b E 4, there exists a, > a and b, > b such that a, V b, = 1 and a, A b, = a A b. Reduction property has roots in descriptive set theory of pointclasses in Polish spaces. Many well known effectivizations of the classic pointclasses enjoy the reduc tion property. Some interesting collections of sets in computability theory satisfy either the reduction property or its dual form, the separation property. Proposition 2.2.4 (Herrmann[unpublished]) For each 117 class P, the lattice n(P) satisfies the dual reduction property. Proof: Let P1 and P2 be (nonempty) 131 subclasses of P and, for i = 1,2, let T, be a computable tree such that P, = [Ti] is the set of infinite paths through Ti. We define computable trees Si such that T Cg S. with Si l s2 = T fn T2 and S1 U S2 = {0,1}<` and with Q, = [Ti]. It will follow that Qi n Q2 = PI n P2 and that Q1 U Q2 = {0,1}. For the first condition suppose that x Qi f Q2, then x[n E i fn S2 for each n, so that x[n E T1 n T2 for each n, and therefore x PI n P2. For each x, we have that for each n, either x[n E St or xrn E S'2. Thus without loss of generality x[n E S1 for infinitely many n. Since S' is a tree, x[n E S'1 + x[m E S'1 for m < n, so that x[fn E S1 for all n and therefore x E Qi. The definition of the trees Si is by recursion on the length of a E {0, 1}<'. First put the empty string in both S1 and S2 since it is in T1 n T2. Now assume by induction that for strings a of length < n, the following hold; aE 1 U S'2 and a E 5si n s2 == a Ti n T2. Now for r = aoO or al, there are four cases: i) If r E T1 n T2, then we put r Si nl S2. ii) If r T1 \ T2, then we put r E S1 \ S2. iii) If r E T2 \ T1, then we put T E S2 \ S1. iv) If r T, U T2, then we consider whether a E S1 or S2. If a E S2 \ S1, then we put r S2 \ S and otherwise, we put r E S \ S2. It follows that in each case, if r E Si, then a E Si, so that each 5, is a tree. The conditions (i) and (ii) follow from the construction by induction on the length of a. The proof ends with the observation that the pair Q1, Q2 dually reduce the pair of P1, P2. 20 The following result, is helpful in determining the isomorphism type of Ln(P), for certain 11' classes. Theorem 2.2.5 (Cenzer and Nies [10]) For any finite distributive lattice L with the dual reduction property, there exists a II class P such that I2 (P) is isomorphic to L. Furthermore, the theory of IC*(P) is decidable. Let P be as in the above theorem for some finite lattice L. The decidability of the theory of L*(P), is a major point of difference between the lattice of c.e. sets, and the lattice of 1l classes. Nies [37] proved that, in the lattice , the theory of each interval which is not a Boolean algebra interprets true arithmetic and is therefore undecidable. However, according to the above theorem, there are initial segments in 4n, of the form [0, P] = Ln(P), which are not Boolean algebras, but with a decidable theory. The decidability of 4n(P) is proved using a result of Lachlan[31], which shows the theory of Ln(P), is manyone reducible to the theory of the finite lattice L*(P). On the other hand, the decidability of P, would imply the undecidability of the theory of LrIn(P). Cenzer and Nies [10] have recently shown that for a decidable II class P, if 4n(P) is not a Boolean algebra then the theory of 4n(P) interprets true arithmetic and is therefore undecidable. Thus we have the following. Theorem 2.2.6 Let P be a II? class such that for some finite lattice L with the dual reduction property, such that C*(P) L, then the tree Tp is not computable, i.e. P is not a decidable IIH class. The observation made in the above two theorems makes the construction of a ll class realizing a finite lattice a challenging task. Corollary 2.2.7 For each finite ordinal n < w there exists a H? class P, such that L(P) has order type n. Proof: Let L = {OL = a, < a2 < ... < an = 1L} be the lattice of order type n. To apply the above CenzerNies theorem, we observe that L satisfies the dual reduction property. For aj, a, E L with ai < aj we have ai A an = ai = ai A aj and a V an = an = IL. Thus there are initial segments of order type n, for any given finite ordinal. The following theorem shows there are initial segments in L* of infinite order type. Build ing on the proof given in Cenzer and Nies [10] we have shown in Cenzer and Riazati [14] that: Theorem 2.2.8 The lattice C* has principal ideals *(P) of order type w + 1. 2.3 The World of Simple 11 Classes Myhill [34] also asked whether there is a maximal c.e. set in the inclusion ordering modulo finite sets. A coinfinite c.e. set A is maximal if there is no c.e. set W such that W n A and W n A are both infinite. Equivalently, A is maximal iff A* is a coatom(maximal element) in *. Soare [48] mentions the importance of maximal c.e. sets for the following reasons: they are coatoms of * and hence the building blocks for more complicated lattices of supersets; they were the ultimate realization of the Post's search for sets with thin complements; their degrees are exactly the high degrees. Maximal c.e. sets are the simplest of the simple sets. Atoms of the lattice *, and classes with at least nearatomic structure in the lattice, turn out to be analogues of the simplelike sets in the lattice of c.e. sets . The fact that dual notions to simple c.e. sets are of low 2nrank, and rather ad hoc distinctions between different notions of simple c.e. sets, would lead to modifications of these dual notions in the lattice fn of IIH? classes. There are characterizations of simple c.e. sets in the lattice *: Table 2.2: Principal filters of S* c.e. Class A the principal filter generated by A A maximal *(A) {0,1} A quasimaximal C*(A) finite A hyperhypersimple *(A) Boolean Algebra A rmaximal *(A) with no complements A without maximal supersets $*(A) dense We study IIH classes analogous to quasimaximal and rmaximal c.e. sets. Princi pal ideals generated by these classes are not in general finite or noncomplemented respectively. We will also consider H classes with finite or noncomplemented CL(P). 2.3.1 Minimal Extensions of 11 Classes Recall that an infinite II class P is minimal if for each II[ class R E Lr(P), either R is finite or P n R is finite. We have observed that, if P is a minimal 110 class then C*(P) {0, 1}. We focus on possible formations of II? classes as finite principal ideals in LC. Minimal classes are the atoms of L and one such possible formation. The definition of minimal IIH? classes has nontrivial extensions to principal filters of Cn: Definition 2.3.1 A IIH? class Q is a minimal extension of a II? class P C Q if for every n? class R E [P, Q], either R n P is finite or Q n R is finite. Remark. If M is a minimal class with M n P infinite, then P U M is a minimal extension of P. On the other hand, adding a copy of any minimal class M to any interval not intersecting P, yields a trivial minimal extension of P which has a new limit path. Thus we make the following. Definition 2.3.2 A H? class P is said to admit Q as a proper minimal extension (P Cmin Q) if Q is a minimal extension of P, and D(P) = D(Q). We now show that certain II? classes have proper minimal extensions. We need to use the limit lemma, that any function f that is computable in 0' has a uniformly computable approximation {f8}s such that lim,+oofs(x) = f(x). Theorem 2.3.3 Each II? class P with a single limit point A with A < 0', admits a proper minimal extension. Proof: Let P and A be as described. Let As be the uniformly computable approximation given by the Limit Lemma (see Soare p. 57). Since A P = [S], we may assume that A'[s E S for all s. If it is not, simply find the longest initial segment a of As [s which is in replace As [s with any extension r of a which is in S and has length s. For any fixed n, there exists an m such that A [n < A3 [s for all s > m and it follows that the modified version of As also extends A [n for all s > m, so that we still get A as the limit of the sequence A3. The minimal extension Q of the class P is obtained by adding to P an infinite sequence B, of new isolated paths such that A [n < Bn for each n. This immediately ensures that Q can have no new limit paths. (If some B E Q is a new limit path, then we must have B(n) 5 A(m) for some least m. Then the interval I(B[m + 1) cannot contain Bn for any n > m and can only contain finitely many of the isolated paths from P, since B was not a limit path of P. But this contradicts the fact that every neighborhood of a limit path must contain infinitely many elements of the class.) This means that Q will be a proper minimal extension of P if we can show that it is a minimal extension. To ensure that Q is in fact a minimal extension, we need to show that for any II class Pe, if P C Pe C Q, then either P, P is finite or Q Pe is finite. To accomplish this, we require that for each e, there exists some k(e) such that if k(e) < i < j and Bi Q, then Bj E Q. Thus if P, P is infinite, then it must contain some Bi with i > k(e) and hence must contain all Bj with j > i. The construction of the tree T is in stages s, using a priority argument. We will define a computable sequence of threshold numbers n(s) and at each stage s, the following are defined. The threshold number n(s) and the tree Ts = T f {0, 1}(S). For i < s, the sapproximation /3O of Bi such that /3' S. T will be a computable tree since for a C {0, 1}', we have a T ==* a E T'. The new isolated paths Bi will be defined by B, = lims,.oo /. There are two types of requirements for the construction. To ensure that As < Bi, we have: Rj: A8[i < Of,: The Requirements: To ensure that for every II class Pe with P C Pe C Q, either Pe \ P is finite or Q \ P, is finite, we have, for each j> i > e, the requirement: R,,i,j: if/37' E T, then /3j E Te. Note that for each e, this requirement only needs to be satisfied for sufficiently large i, that is, for i > k(e), where k(e) is a function computable in 0', but not necessarily computable, which will be shown to exist later. Priority is assigned to the requirements as follows. Rj has priority over Rk if j < k. Rk has priority over Re,i,j if k < i. Re,i,j has priority over Re',i',j' if either (1) 1<' or (2) i = i' andj at stage s + 1 when /O3 does not extend AS+'1 [i. Before describing the action to be taken for this requirement we note that since P = [S] has only one limit path, every node a E S has an extension which is not in S. The action for this and the other requirements are all going to require defining nodes oa, and 7, as follows. Let a, be the shortest and then lexicographically least extension of As+l [n(s)(1 As+1(n(s))) which is not in S and let y, be the shortest and then lexicographically least extension of As+l fn(s) + 1 which is not in S. We are always going to define n(s + 1) to be the maximum of {f II, IIII}. The action to be taken when Ri requires attention is the following. We need to redefine /3 and also define /3 for the first time. Define f+l to have length n(s + 1) and extend a, by a string of '0's and similarly define 13+ to have length n(s + 1) and extend y, by a string of '0's. For each j < s different from i, let/3J+1 have length n(s + 1) and extend /3 by a string of '0's. The tree TS+l contains all nodes from S of length n(s + 1) as well as the nodes /3+' for all k < s. Requirement Re,i,j requires attention at stage s + 1 when (i) As+'ri < Bj3O, (ii) 1js0 ( Te and 03'O E Te, and (iii) for all d < e, if /3'0 Td, then /3;O0 Td. The action to be taken when Re,i,j requires attention is the following. We let /3s+ be the sequence of length n(s + 1) which extends /3j&0 by a string of '0's, we let O'+1 be the sequence of length n(s + 1) which extends a, by a string of '0's and we let /3+1 be the sequence of length n(s + 1) which extends 7, by a string of '0's. For each k < s different from i and j, let 3k+1 have length n(s + 1) and extend O by a string of '0's. The tree TS+l is defined as above to contain all nodes from S of length n(s + 1) as well as the nodes /O+1 for all k < s. Initially, we let n(0) = 0 and let To = {0}. At stage 1, we have TV = {0, (0), (1)} and we have 0 = (1 A'(0)). At stage s + 1 > 1, we take action on requirement of highest priority which needs attention. If no requirement needs attention, we just let n(s+1) = Ia,, let /3s+1 = a, and for each j < s, let /3j+' have length n(s + 1) and extend /J by a string of '0's. The tree TS+' is defined as above. It is clear from the construction that for each s, /O,/,... ,/3 and As[n(s) are all distinct nodes of length n(s) in TP. We have to show that each requirement is eventually satisfied and that for each s, lims)co3 = B exists and belongs to Q = [T]. A key concept in showing this convergence is the notion of the estate of a finite or infinite path. An infinite path B has estate (co, c1,...) where Ce = 1 if B E Pe and c, = 0 otherwise. For the finite path /3, we define the estate to be (Co, ci,..., c,) where ce = 1 if and only if 3 E Te. In either definition, the estates are ordered lexicographically, so that (co,..., c,) is lower than (cr,..., c) if ce least such that they are different. An important observation is that whenever we take action on any requirement Re,i,j at stage s + 1 > i, we lower the estate of 3, that is, the estate of f0+1 is lower than the estate of /3f. The action taken change Ce itself from 1 to 0 and, for any d < e, Cd can only decrease because of the final clause in the definition of requiring attention. On the other hand, if we take some other action at stage s + 1, then the estate is either the same or lower, since 03+1 will then be an extension of f, so that /3+1 Te implies O?, E Te for any e. This demonstrates the following claim. CLAIM 1: For each i, the estate of /3i converges. The next step in showing that the construction converges is the following. CLAIM 2: Each requirement only requires attention at a finite number of stages. Proof of Claim 2: This Claim is proved by induction on the priority. Suppose that all higher priority requirements have been satisfied, so that our requirement has highest priority from some stage s on. There are two cases. For the requirement Ri, we may assume that s > i and that s is large enough so that At[i = A[i for all t > s. If Ri ever requires attention at some stage t > s, then we take action and get A[i < /0'. The only other action which could affect this requirement is action on some Re,i,j for some j > i. But this action will make Oi'3+ an extension of Ar+1 [n(s) and hence still an extension of A[i. For the requirement Re,i,j, we simply take a stage s large enough such that the estate of 3i has converged and all lower priority requirements have ceased requiring attention. If Re,i,j ever required attention after stage s, we would take action and thus lower the estate of /03, a contradiction. Once the requirement Ri and the estate of fh converge, then Bi can converge. CLAIM 3: For each i, the sequence /3f converges to an infinite path B, E Q. Proof of Claim 3: By the previous claims, we may take s large enough so that the estage of /3 has converged and so that Ri no longer requires attention. Then after stage s, any action taken only extends /3f by a string of '0's. Thus Bi = 3'0w. Now we can determine the structure of our Hl class Q = [T]. CLAIM 4: Q=PU{Bn : n < w}. Proof of Claim 4: To see that each Bn is in Q, take s such that B. = /,'0,w for all t > s. Then for any t > s, Bn [n(t) = /3 Tt, so that B, [T] = Q. Now consider an arbitrary B Q. For each s, B[n(s) E Ts, and is thus either in S or equal to some #3. If the former happens infinitely often, then B E P, thus we may assume without loss of generality that Brn(s) = '3s,) for some sequence i(s). Now if there is a fixed I such that i(s) = i for infinitely many s, then B = Bi, as desired. Otherwise, there must be infinitely many stages s+1 such that i = i(s+l1) 5 j = i(s). But this means that /3? < /3+1, and this can only happen when we act on some requirement Re,i,j with e < i < j. But this makes i(s + 1) < i(s), which can only happen finitely often. Next we check that the B, will approach A in the limit. CLAIM 5: For each i, A[i < Bi. Proof of Claim 5: By the previous claim, Bi = ,/jO, where s is large enough so that R, never requires attention after stage s and by the argument in Claim 2, large enough so that A"[i = A i. It follows that Af[i = As [I < Bi. We now consider a stronger notion of convergence of the estates, necessary to obtain the minimality condition. CLAIM 6: The estates of B, converge to a limit. Proof of Claim 6: This means that for each e, there exists some k(e) such that for anyj>i> e, Bi Pe P, : Bj EP. Definition of k(0): Case I: For all k, Bk i Po. Then let k(0) = 0. Case II: There exists k such that Bk E P0. Then let k(0) be the least such k. Now suppose that j > k and consider a stage s large enough so that the 0states of both Bj and Bk have converged and such that Bj = j0syw and Bk = /,30w and suppose by way of contradiction that Bj Pe. Then A[j < /3yjO and we have (3'0 E To and f#j0 To, so that requirement Ro,k,j would need attention and we would act to put f0+' To, contradicting our assumption of convergence. Definition of k(e + 1): Case I: For all k > k(e), Bk i Pe+i. Then let k(e + 1) = k(e). Case II: For some k >_ k(e), Bk E Pe+i. Then let k(e + 1) be the least such k. Now suppose that j > k and consider a stage s large enough so that the e + 1states of both Bj and Bk have converged and such that Bj = 3J0w and Bk = 0I0 and suppose by way of contradiction that Bj ( Pe+i. First observe that we have /3#'0 E To and /30 (O To. Next let d < e and suppose that Pj'O i Td. Then Bj Pd and j > I > k(e + 1) _> k(e), so that Bk Pd either, which means that Ok'0 Te. Thus requirement Re+,k,j would need attention and we would act to put k+1 T,+1, contradicting our assumption of convergence. Finally, we can demonstrate that Q is a minimal extension of P. CLAIM 7: For any e, if P C Pe C Q, then either Pe P is finite or Q Pe is finite. Proof of Claim 7: Suppose that Pe P is infinite and let k(e) be given by Claim 6 so that for m > n >_ k(e), Bm P, if and only if B,, E Pe. Then there is some n > k(e) such that Bn E Pe and therefore Bm E P, for all m > n. Thus Q P, is infinite as desired. CLAIM 8: D(P)= D(Q). Proof of Claim 8: Let A be a limit point of Q which is not a limit point of P. First suppose that A ^ P. Then (since P is closed) there is a clopen set U with A E U such that P n U = 0. Then Q fl U is a H1 subclass of Q which is disjoint from P and contains A. Since A is a limit point of Q, Q n U must be infinite. Now consider the IIH class R = P U (Q n U). Since R \ P is infinite, it follows from the definition of minimal extension that Q \ R = (Q \ U) \ P is finite. We can obtain a [ class M disjoint from P with Q = P U M, as follows, let (Q \ U) \ P = {B, ..Bt}, for each i, choose a clopen set Vi with Bi Vi and Vi n P = 0 and let M = (Vi U... U Vk U U) n Q . If M is not minimal, then clearly P U M is not a minimal extension of P. If M is minimal, then P U M is not a proper minimal extension. Next suppose that A E P but is isolated. Then there is a clopen set V with A E V such that Pf V = {A}. But Q n V is infinite, since A is a limit point of Q. It follows as above that (Q \ V) \ P is finite, and once again we can define a 11' class M, disjoint from P with Q = P U M This completes the proof of Theorem. Remark. If P Cmin Q then [P, Q]* {0,1}, the trivial Boolean algebra. Remark. Let Q be a minimal extension of P, such that Q 5 M U P for any minimal II class M. Then Q is a proper minimal extension. Theorem 2.3.4 Let P be an infinite II class. Then P does not admit a decidable proper minimal extension. Proof: This follows from our splitting theorem 2.5.3 below. Given P C Q with Q a decidable 11 class, we consider the following two cases: Case I: If P is complemented, then Q \ P is a 11 class and Q = P U (Q \ P) thus Q \ P must be minimal and Q is not a proper minimal extension of P. Case II: If P is not complemented, then by splitting theorem there exists a 11' class P1 with P C PI C Q with P \ P and Q \ P1 both infinite, so that Q is not a minimal extension of P. Remark. This implies that structure of decidable classes is much simpler. 2.3.2 Quasiminimal H11 Classes Definition 2.3.5 Let M =< Mo, M1, ..., Mi > be a finite sequence of H1 classes. If Mo is a minimal class, and Mi Cmin, M+l, Then M is called a minimal chain, of length n. Definition 2.3.6 A 117 class P is called a pure extension of a class M, if for some minimal chain < Mi : i < n >, M = Mo and, P = Mn. We will refer to terms of arbitrary minimal chains as pure Ho classes. Lemma 2.3.7 For each R11 pure class Q, the lattice C(P) is finite. Proof: Since Q is pure, then Q is the union of a minimal chain M1,..., Mn = Q. We show by induction that LC(Q) is finite. For n = 0, M0 is a minimal class, and thus C(Mo) has two elements. Now suppose that IL(Mk) is finite with < 2k+l elements and Mk+j is a minimal extension of Mk. Let P be a 117 subclass of Mk+m, and let P1 = P n Mk C Mk. There are at most 2k+l choices for P1, modulo finite difference. Now we claim that for each P1 C Mk there are at most 2 subclasses P of Mk+j with P nl Mk = P1, so that card(C(Mk+i) < 2k+. Given P and P1 = P fl Mk, let P+ = P U Mk, then Mk C P+ C Mk+i, so there are two cases. First, P+ \ Mk could be finite, in which case P C* Mk and P = P1. Second, Mk+m \ P+ could be finite, in which case P =* P1 U (Mk+i \ Mk). In general Mk+j \ Mk may not be a 117 class, so we can only say that card(I (Q)) < 2k. Definition 2.3.8 A 11 class Q is said to be a linear extension of a 11' class P, if the interval [P, Q] is linearly ordered. Observe that Q is a linear extension of P if and only if there do not exist distinct II? classes P1, P2, P3 in [P, Q]* with P1 =* P2 U P3, and P2 incomparable with P3 modulo finite differences. Remark. Not all minimal extensions are linear. Recall the 5element lattice. {0, Po, Q1, Q2, P1} where Qi and Q2 are complements in [Po, Pf1], but < 0, Po, Qi, P1 > is still a minimal chain. This example shows that finiteness of L(P) does not imply that P = U{M1, M2, ..., Mn}, for finitely many minimal II classes M1, ..., Mn. While in the lattice of c.e. sets modulo finite differences, S*, finiteness of *(A), a principal filter of $*, would impose an antichain structure on c.e.supersets of A, moreover A would have to be the intersection of finitely many maximal c.e. sets. This example shows that we need the following definition. Definition 2.3.9 A 11 class P is quasiminimal if P is the union of finitely many pure H' classes. Remark. A decidable quasiminimal H' class is a union of disjoint minimal classes. This follows from theorem 2.3.7 Theorem 2.3.10 A H' class P is a finite union of minimal classes if and only if 12(P) is a finite Boolean algebra. Proof: Suppose first that P = Pi U P2 U ... U Pn is a finite union of minimal II? classes. We may suppose without loss of generality that P, : Pj for i j, since if Pi n P1 is not finite then Pi =* P, n P, =* Pj and Pi U P, would also be minimal. Now if Q is a 1 subclass of P, let F = {i : P, n Q is infinite }. Since each P, is minimal, Pi n Q =* P, for i F and of course Pj n Q =* 0 fori F. Thus Q =* UiEF Pi and has complement Q =* UjFPj in L (P). On the other hand, suppose that L(P) is a finite Boolean algebra and let P1, P2, ..., P, be the atoms. It is immediate that P =* PI U... U Pn. Now Pi U... U Pn = Po C P and we may obtain P = Po as follows. Let P \ Po = {AI, ..., Ak}, for each i, take an interval Ui with Ai E Ui and Ui n Po = 0 and let U = U1 U ... U Uk. Then P n U = {AI, ..., Ak}. Now simply adjoin this l' class P n U to P1. Theorem 2.3.11 For each H1 class P, P is a quasiminimal HI class if and only if I(P) is finite. Proof: First suppose P is a quasiminimal, that is P = Q1 U ... U Qn, with each Qi a pure IH' class. By the lemma, for each i, L*(Qi) is finite, say ki many elements. Then any subclass Q of P can be written as Q = (Q n Qi) U ... U (Q n Q,n). Since there are < ki many choices for each i this shows that card( (Q)) <_ k, x k2 x ... x k, and is therefore finite. Now suppose that C*(P) is finite, we prove by induction that P is quasiminimal. If Card(IC(P)) = 2, then P is minimal. Next suppose that Card(IC(P)) = n + 1, and let R1, R2,..., Rk be coatoms in C*(P). Then for each i, P is a minimal extension of R, and, by induction (Since Card(IC(Ri)) < n), each Ri is the union of some minimal chain, Mij, Mi,2, ..., Mi,k(i). Then Mi,, ..., Mi,k(i), P is again a minimal chain and P is the union of those minimal chains. Corollary 2.3.12 For any H' class P, if C*(P) is finite, then L(P) is a Boolean algebra. Definition 2.3.13 For each quasiminimal IH? class P, the quasirank of P is defined to be the cardinality of the lattice C(P). Table 2.3: Structure of principal ideals of Cn II Class P The Principal Ideal Generated by P P minimal C(P) 0 {0,1} P is quasiminimal C(P) finite P is finite union of minimals *(P) finite Boolean algebra P thin Cn(P) Boolean Algebra Non pure quasiminimal 11 classes are interesting for their splitting properties. We will elaborate on this and other splitting properties in what follows. 2.3.3 Fully Complemented Principal Ideals In E, the notion of hyperhypersimple c.e. sets can be singled out via different approaches. The Owings Splitting Theorem implies that the principal filter generated by a hyperhypersimple c.e. set is fully complemented (a Boolean algebra) and vice versa. Clearly, a coinfinite c.e. set is hyperhypersimple if and only if for every c.e. set B, if A C B then A U B3 is c.e.. Definition 2.3.14 Splitting property for c.e. sets. A c.e. set S has the splitting property if every c.e. nonrecursive set A can be split into two nonrecursive c.e. sets B and C such that B C S. Remark. Hyperhypersimple sets have the splitting property. Remark. The c.e. sets with the splitting property form a filter in C. Remark. The possible isomorphism types of C*(A) for A hyperhypersimple are exactly the E' Boolean algebras. Lemma 2.3.15 The theory of hyperhypersimple sets is decidable. Proof: Since the lattice of all hyperhypersimple sets modulo finite sets is relatively com plemented, by the same proof which shows that C*(A) is a Boolean algebra for A hyperhypersimple, the result then follows from Ershov [18]. Definition 2.3.16 A Ho class P is thin if for each Hl class Q E n(P), there is a clopen set U C 2W such that Q = U n P. The principal ideal generated by a thin H' class P is a Boolean algebra. The first example of a thin II class is due implicitly to D. Martin and M. Pourel in [33]. They constructed an axiomatizable, essentially undecidable theory T such that every axiomatizable extension of T is finitely axiomatizable over T. It is easy to see that the class of complete extensions of such a theory T is a thin II? class, and it is perfect because it contains no computable element. The first explicit example of a thin IIH? class is due to S. Simpson(unpublished). Countable thin HI? classes of arbitrary rank, including minimal classes, were con structed in Cenzer et. al. [7]. Remark. If P is a thin 1? class if and only h(P) is a Boolean algebra. Lemma 2.3.17 (Cenzer et al. [7]) For any thin II? class P and any element x of P, x is isolated in P if and only if x is computable. Lemma 2.3.18 (Cenzer et al. [7]) Let P be a IIH? class. i) If P is thin, and D(P) is a singleton, then P is minimal. ii) If P is minimal and infinite, then D(P) is a singleton. Theorem 2.3.19 (Cenzer et al. []) Existence of Thin Classes. For every recursive ordinal a, there is a thin II? class P, with CantorBendixson rank a, moreover P, may be taken to be the set of paths through a computable tree with no dead ends. Perfect thin IIH? classes were constructed by Simpson and are related to supermin imal profinite groups by the work of R. Smith [46]. If P is a perfect thin IIH' class with 0 = Q C P then Q is also perfect thin. Each such class can be split into two perfect thin classes by a clopen set. Let P be a minimal and thin II? class. Then D(P) = {A}, and A is non computable. If B E D(P), and P a thin I? class, then B has to be non computable. Therefore the notion of CantorBendixson rank happens to capture the computational complexity of members of a thin IIH? class. On the other hand minimal classes (that are forced to have unique limit path) with computable limit path are not thin, and thus generate a rich structure as their principal ideal, as we shall see below. Indexing H1 classes, corresponds to a fixed enumeration of the c.e. sets. Definition 2.3.20 A Hl class P is said to be the eth HO class, i.e. P = PF, if and only if, x E Pe = (Vn)(xan r We). Index sets for IIH classes were developed in CenzerRemmel [12]. It is important that we can express Pe uniformly as the class of infinite paths through a computable tree Te. Lemma 2.3.21 There is a computable relation R C w x 2< such that, for each e, the set T, = {: R(e,)} is a tree and Pe = [Te]. Proof: We have x E Pe == (Vn)(x[rn we). Let R(e,a) <= (Vr < o)(r ( We). First we check that Te as defined above will be a tree. Suppose a E Te and al < a. Then for any r < oa, we also have r < a so r We. Thus ai E Te. Next we check that x E PFe == (Vn)xrn E Te. Suppose x E Pe and let a = xrn,r < ao. Then r = xrm for some m < n, so that T we. Thus xrn E Te. Suppose next that (Vn)x[n E Te, then certainly for each n, xrn V We, so that x E Pe. The complexity of local complementation in n, C*, is used to find the isomor phism type of these structures. Arslanov [1] proved there is no coinfinite c.e. set A such that the filter C(A) is effectively complemented. Similar results holds in Cn(P). Theorem 2.3.22 There is no infinite I1 class P such that Cn(P) is effectively com plemented, i.e. such that, for some computable function f, P. C P => P () U P, = P A Pf() n PF = 0 Proof: Let Ph(.) = P, f P. Then fh is a computable function, and hence it has a fixed point z for which P h(z) = P,, so that P, is a complement of P n P, a contradiction. Arslanov et. al. [2] showed that there is no coinfinite c.e. set A, such that, for some function f : Theorem 2.3.23 There is no infinite H1 class P s.t. for some function f :T )C, the principal ideal LC*(P) is fcomplemented, i.e. P C P = Pf(,) U P =* P A Pf(,)nP =* 0. Proof: For each set X, cx be applied to show that iff such that Wf(.) =* W,. The rest is similar to the case of Cn(P) . Thus for each H? class Q E n(P) the complement of Q in CL(P) is not even computable in the complete c.e. set C. Possible isomorphism types of LCn(P) for P thin, can be contrasted with the prin cipal filter generated by a hyperhypersimple c.e. set. Lachlan [31] has proved that the possible isomorphism types of C*(A) the principal filter generated by a hyperhy persimple c.e. set, are exactly the E0 Boolean algebras. The c.e. sets are closed under union and intersection uniformly effectively, namely there are computable functions f and g such that Wf(,,y) = Wx U Wy, and Wg( ,y) = W, n Wy. Also one might observe that Wx C* Wy is Ecomplete in x and y, (Wy is cofinite = W C* Wy) also W_ C Wy is IIcomplete, making it necessary for the Boolean algebra to be at least E0. The lattice operations n and U are also computable in r. In general the 37 relation P, C Py is also 11' complete and P, C* Py is also E complete. However for a thin 11 class, n(P) has a simpler representation. Theorem 2.3.24 For each [? class P, If P is thin then the lattice Cn(P) is a c.e. Boolean algebra. Proof: The Boolean algebra L(P) may be realized as a quotient of the Boolean algebra of clopen subsets of 2' intersected with the H class P. In other words, for Q, R as above, there are clopen sets U, V such that Q= P N U, R = P n V. For each pair of clopen sets U, V, let U =p V <=' U fn P = V n P. First observe that for U = I(oai) U ... U I(ak), U n P = 0 4== a, Ext(P) A ... A ak i Ext(P) which is a E condition on U. Then u n P C V n P == (U\V) n P =0, and (UnP=VnP) #= 7UnPCVnPandVnPCU nP. Thus thepartialorder and the equivalence relation are both E. Remark. The lattice C(P) is certainly a E' Boolean algebra, since "U n Pe is finite "is E and U n P C* V n P 4= (U \ V) n P is finite. Since the relation "U N P, is finite is XOcomplete, it seems that C(P) should not in general be a So Boolean algebra. However, we can still show that for a thin class P, n(P) has a complement function computable in 0". Theorem 2.3.25 For each thin H class P, there is a function f Pr C P = Pf(x) U Px = PandPf(x) n P = 0. Proof: We observe that the condition P, n Py = 0 is 1H and the condition P, U Py = P is II0, so that both can be verified using a 0"oracle. Given that P is thin, r(P) is complemented, so there exists for every x, a Pcomplement Py for P, such that (P, nPy)U P = P and Pyn(P, nP) = 0. Define f(x) to be the least y which satisfies the two conditions. Definition 2.3.26 A II? class Q is a thin extension of P if for each R E [P,Q], there is a clopen set U such that R = PU (Q n U) = Q n (Pu U). Remark. Our example of a non pure quasiminimal IIH class would show that, thinness is not necessarily equivalent to, [P, Q] being complemented, that is, for any R E [P, Q], there exists some S E [P, Q] such that R n S = P and R U S = Q. Clearly if M is a thin class with M \ P infinite, then P U M is a thin extension of P. Thus we say that Q is a proper thin extension of P if in addition there is no thin class R such that Q = R U P. Remark. The existence of proper thin extensions follows from that of minimal extensions. 2.3.4 Where There Are No Complements; Nerode's Theorem The opposite notion to that of thin IIH classes, is a II class P where no non trivial element of C(P) is complemented. The above example of a non pure quasiminimal class as the join of two proper minimal extensions, is one such II class, for which C*(P) has 5 elements, 0, a minimal class M, two of its minimal extensions, Qi and Q2 and Q1 u Q2. Definition 2.3.27 An infinite IIH? class P is 7rminimal if CI(P) has no non trivial complemented elements. Our terminology is similar to that of rmaximal c.e. sets. An rmaximal c.e. set A has an rcohesive complement, i.e. A is infinite and can not be split into two infinite parts by a recursive set. Let us recall equivalent descriptions of rmaximal c.e. sets; Lemma 2.3.28 For each c.e. set A, the following are equivalent; i) The filter Ce*(A) has no nontrivial complemented elements. ii) A is rcohesive. iii) A is infinite, and there is no pair of c.e. sets whose union is w and whose intersection is A. Lemma 2.3.29 (Robinson[42]) There exists an rmaximal set which is not maximal. The following theorem of Nerode characterizes the nonprincipal ultrafilters of the Boolean algebra of computable sets, REC*, as the intersections of REC* with the principal filters of P(w) (modulo the finite sets) generated by the recursively indecomposable sets: Theorem 2.3.30 (Nerode, Shore [36]) The non principal ultrafilters of REC* are exactly those of the form UEC = {X* : A C* X E REC}, where A is an rcohesive set. Or equivalently A is an rcohesive subset of w = {X* : A C* X E REC} is an ultrafilter of REC*. Nerode's theorem suggests similar characterizations for the notion of 7rcohesiveness in the lattice C*. Definition 2.3.31 Let .F C Ln, for each fl' class P, the collection V) = {Q* : P C* Q E F} is called the .Fcovering filter of P. Definition 2.3.32 An infinite Ho class P, is said to be Kcohesive if for every clopen set U, either U fn P is finite or U nf P is finite. Lemma 2.3.33 For each flo class P. If P is 7rminimal, then P is Kcohesive. Proof: Suppose P is not Kcohesive. Choose a clopen set U such that both U n P and U n P are infinite. But U n P and U n P are both fl' classes and provide non trivial complemented elements of *(P). In other words, for 7rminimality, if we let FT be taken to be the clopen subsets of 2', unfortunately such a natural analogy fails to capture all 7rminimal classes. In fact in C: Theorem 2.3.34 For every IIH class P the following are equivalent: i) P is Kcohesive. ii) {K: P C* K E /C(2w)} is an ultrafilter. iii) P is a 11' class with a unique limit point. Proof: i) + ii) Assume that P is Kcohesive. the collection V1p is clearly a filter for any P. Let the clopen set U be given. If U n P is finite then P C* U so U E V^p. If U l P is finite, then P C* U so U Vcp . ii) + iii) Suppose by way of contradiction that P has 2 limit points A, B. choose a clopen set U so A E U and B U. Then U l P is infinite, since it has a limit point A, and likewise U n P is infinite. iii) + i) Assume that P has a unique limit point A and let U be a clopen set. Without loss of generality suppose A E U. Then U nl P has no limit point and is therefore finite. The result of Nerode, suggests a different way to characterize nonprincipal ultra filters of C, or at least a new type of ultrafilter of L. This follows from switching from clopen indecomposability to decidable indecomposability. Recall that, a 11 class P is said to be decidable, if P = [T], for some computable tree T without deadends. The collection of all decidable IIH classes is denoted by DEC. The lattice of LDEC modulo the ideal of finite H' classes will also be denoted by CDEC*. 2.4 Principal Covering Filters The notion of an ultrafilter is suitable for the Boolean algebra of REC*, but needs to be weakened for the lattices Cn, ECE,* "CDEC*, which are not complemented. However the notion of a filter and its closure properties still makes sense. But we will replace the notion of ultrafilter with the notion of prime filter. Definition 2.4.1 For each FT C TP(2), and each l1 class P, a filter of the form VPF = {Q* : P C* Q E F} is called Prime if for each pair of fH classes Q, R E FT, Q U R Vp implies that Q E Vr or R E V. Remark. If Vp is prime, then P is 7rminimal, since a pair of complements Q, R in L*(P) would violate the conclusion of the primality. However, the II class with the 5element lattice, discussed above is irminimal, but Vp is not prime, since Qi U Q2 E Vp, but neither Qi nor Q2 is in the filter VpF. Definition 2.4.2 A IIH class P is called JCohesive if the filter VDEC = {Q* : P C* Q E DEC} is a prime filter of DEC*. Lemma 2.4.3 For each H? class P, VDEC C V'. If P is 8cohesive then P is K cohesive. Theorem 2.4.4 For each H1 class P: i) If VDEC is prime then Vp> is prime. ii) If V~p is prime then for some A E P, D(P) = {A}. iii) If VJDEC is prime then CEC*(P) has no nontrivial complemented elements. Proof: All clopen sets are decidable 11' classes, this proves i). To see ii) we observe that for the Boolean algebra KA(2w), prime implies ultrafilter, so the previous theorem applies. For the last part, If Vp is prime then P can not be split by elements of Jr, that is we can not have fl' classes P and Q in F such that P f Q and P n R are non trivial complemented elements in C*(P). (Since we would have P C* Q U R but neither P C* Q nor P C* R.) Corollary 2.4.5 For each H' class P; If Card(D(P)) > 2 then VDEC is not prime. Theorem 2.4.6 For each fl class P, with D(P) = {A}. If the filter Vp is not prime then A is computable, and the filter VDEC is not prime. Proof: 1 00 Since Vp0 not prime, we can assume that P admits a splitting into infinite IIr Hio classes Q1,Q2 with Q1, Q2 VP'. Thus P C* Qi U Q2 and Qi n Q2 =* 0. We can modify Q1, Q2 to get exactly P = Qi U Q2 and Q, n Q2 = {A}. For the first part we get Q1 U Q2 C P, by replacing Qi with P n Qt. To get P C Qi U Q2, consider the finite set F = {A1, A2,..., Ak} = P \ (QI U Q2). These are all isolated in P, and therefore computable. So F is a IIH class. Now replace Qi with QI U F. Each infinite 11H class P fl Qi, i E {1, 2} has to contain some limit point, and that has to be the unique limit point of their superclass P. Thus A E Qi n Q2. But the Hl class Q fn Q2 is finite and every element of a finite I1 class is computable. We obtain Qi n q2 = {A} as follows. Let D = Qi n Q2 \ {A} = {Bf1, B2,..., Bm}. Each B, is isolated in P, so we can get a clopen set U such that U n P = D. Now replace Q2 with Q2 fl U. Now suppose there are computable trees S1i, S2, such that Qi = [5'i],Q2 = [S2], and P = Qi U Q2 and Qi n Q2 = {A}. We use a technique from the E1 separation theorem, to construct R1, R2 E CDEC* such that P = R1 U R2, with R1 n R2 = {A}, and moreover, Qi c Ri, Q2 C R2. We will define decidable trees TI1, T2, such that R, = [T1] and R2 = [T2]. For a < A, we put 0a E T1 U T2. For every node oa A, we must decide if a E Ext(Ri) = T1, or a E Ext(R2) = T2 or none. Let p cr with pl = n such that p < A, and p'ao(n) : A(n), (where ar branches off the limit path). Now by the assumption Qi n Q2 n I(pa(n)) = 0, so by Konig's lemma there exists I such that poa(n) has no extension of length I in si 5'2. Compute the least such I and decide a as follows. 43 First suppose 1o, < I. If a has an extension of length 1 in Si we put ao in T1. If c has an extension of length I in S2 we put a in T2. Next suppose al7 > 1. Then for each i, we put a E Ti, only if aol E Si. Clearly T1 and T2 are decidable trees, and R1 n R2 = {A}. Let us check that Qi C Ri. Suppose x :7 A and x E Qi. Then by assumption x Qii. Let p be the largest common initial segment of x and A and let I be the least such that px(n) has no extension of length I in S1 n S2. Then x[l E Si, hence x[l E T, and therefore x E Ri. Remark. If P is decidable with P = [T], then we can modify the definition of T, and T2 to make P = R1 U R2 as follows. For the case when Io > I just put or E Ti <=' arI E Si and 0r E T. This remark essentially proves that: Theorem 2.4.7 For each decidable II? class P, with D(P) = {A}, if P is 7rminimal no then Vpo is prime. Remark. If P is a minimal and thin II? class, then C*(P) is just the two element no Boolean algebra so Vp1 is prime, and D(P) = {A}. When the limit point is not computable, non thin minimal classes, and non min imal thin classes, have non trivial complemented principal ideals. The following theorem of Lachlan [31] relates three notions of rmaximal c.e. sets of integers, the hyperhypersimple c.e. sets, and maximal c.e. sets. Theorem 2.4.8 (Lachlan [31]) A c.e. set is maximal if and only if it is both hyperhypersimple and rmaximal. This suggests that the corresponding notion to rmaximality, namely the notion of 7rminimality, should be intimately related to the computability of the limit paths. Or at least in the case of minimal II classes. This is the case at least for decidable II classes, as we shall see in the following theorem. The following theorem is a lattice theoretic characterization for non computability of the limit path of a decidable II? class. Theorem 2.4.9 Let P be a ?l class with D(P) = {A}. Then no, i) The filter Vpo is prime == A is not computable. no ii) For P decidable, and prime Vp1, if L(P) is non trivial, then C*(P) is infinite. Proof: no For the first part, if the filter Vp1 is not prime, then A is computable by the above theorem. On the other hand, suppose that A is computable. Then {n : A[nU(1 A(n)) V Ext(T)} is a computably enumerable set and must be infinite since A E D(P). Enumerate E as ol, 72,..., and let P1 = P \ UnaI(o2n+1). Then P C P1 U P2, but P \ P1 and P \ P2 are both infinite. Ho For ii) suppose that the filter Vp1 is prime, and that C*(P) is not trivial, and let P0 be a proper II subclass of Po. Thus of course Po has no complement in CI(P), so by the Splitting Theorem L*(P) is infinite. The above theorem suggests the importance of the following II classes. Definition 2.4.10 A H class P is said to be CBcomputable if the set D(P) is computable. Hence, finite and non thin minimal ll' classes are all CBcomputable. And in the light of the above theorem, some irminimal [1' classes are not CBcomputable 1H' classes. 2.5 The Splitting Property The fact that each infinite computable set of integers splits into two infinite com putable sets is rather trivial, but useful in characterizing the Boolean algebra of computable sets up to isomorphism. Theorem 2.5.1 (FreidbergMuchnik Splitting Theorem) Every noncomplemented A E splits into two disjoint noncomplemented c.e. sets B, C e such that A= B U C. This theorem has extensions to nontrivial principal filters (A) of : Theorem 2.5.2 (Owings Splitting Theorem) For any c.e. set D, every noncomplemented element of E(D) can be split into two noncomplemented elements. That is given a c.e. set A such that A U D is not c.e., there are disjoint c.e. sets B and C such that ; A = B U C, and B U D and C U D are not c.e. Owings splitting theorem may be compared to the following result. Theorem 2.5.3 Splitting Theorem for Decidable IIH classes For each noncomplemented Po E n(P), where P CEC, there exists Pi E [P0, P] such that P \ PI and P1 \ Po are both infinite, and furthermore Pi is non complemented. Proof: Let P = [T], Po = [To] for some computable tree T with no dead ends, such that Po has no complements in Cn(P) (or in C*(P)), and so P \ Po is infinite. Let the infinite sequence {o, rl, T2, ...} enumerate the computably enumerable set T\Ext(To). Define by recursion the new sequence {an : n G w} such that : 0`O := TO; an+, := Tj where j = pim.[(m > i) A (on = ri) A (rj {lao, a U,...,0n})]. (That is, an+, is incompatible with oi for each i < n). CLAIM: Card({am :m E w})= No. In other words the sequence never becomes stagnant, i.e. there is always new proper choices for Tj in the above construction. PROOF of the CLAIM: If {am : m E w} = {ao, ..., On}, then every element of T \ Ext(To) is compatible with one of {0, oo, ..., on}. For each k < n, there are just finitely many predecessors to each oan. Let A C {o, ..., n, ...} be the finite set consisting of all such predecessors, namely A = {rj : (30 E {ao, ..., an})(rj < o)}. Let J = A U {aO, ..., rJ, For each x E P \ Po there exists a Tj such that rj < x and Tj is compatible with one of the {ao,i, ..., 0n}. Then there are two cases: i) rj < Ok for some k, 0 < k < n, which implies (rj A) A (x E I(Tj)). ii) Ok Tj for some k, 0 < k < n, which implies that x E I(Uk). Therefore P \ Po = P n K, where K = UJ .7I(p) is a clopen set in C(2w). But this would imply that P n K is a complement of Po in Cn(P), which ends the proof of the claim. Now consider the infinite computable set of incompatible elements: S= {o, ..., On, ...} which is a subcollection of T \ Ext(To). Let us let Pi = P \ UJnE I(U2n). Then P1 is a 11' class. We observe that for each n, I(orn) contains an element of P \ Po. In fact, since T is decidable and the sequence {1, : n G W} is computable there exists a uniformly computable sequence {Xn : n E w} of elements of P \ Po, with X, C I(aE). Moreover, it follows from the incompatibility of aUm and o,, that rn n = Xm : Xn. Thus PI \ Po contains infinitely many elements X1,X3, XS,.... On the other hand P \ P1 is infinite since it contains Xo, X2, X4, .... Let P2 = P \ UI(a2n+i), and observe that P = P1 U P2 and P1 n P2 = Po. Now if P1 had complement Q, so that P1 U Q, = P P1 n Q, = 0, and P2 had complement Q2, then Po = P1 n P2 would have complement Q1 U Q2, thus at least one of P1 P2 has no complement. Corollary 2.5.4 If P E 'EC, and P is not thin, then L(P) is infinite. Proof: Let P be a decidable class with a proper subclass Po which is not complemented. By the splitting theorem, there is a subclass P1 with Po C* P1 C* P with PI also not complemented. Applying the theorem again, we get P2 between P1 and P and by repeating the process, we get Po C P1 C P2 C ... C P, with each inclusion proper. 47 Proof: (alternatively) We can embed a copy of * in [P0, P]*, by taking Qe = P \ UnfEWI(o,,). This map will be discussed in the next chapter. Restating this corollary, we can say that for any decidable IH class P, if C(P) is finite, then LC(P) is a Boolean algebra. This was shown by Cenzer and Nies by a different method. CHAPTER 3 GLOBAL PROPERTIES OF I0 CLASSES 3.1 Definability, and Automorphisms Definition 3.1.1 A property 0 of II classes is said to be invariant in n if and only if it is invariant under all automorphisms of Cn. Definition 3.1.2 A property ?p is called definable if there is a formula of one free variable (with no parameters) in the language of lattice theory which defines the class those classes in Ln having the property 4, with the usual interpretations for relation symbols. Since every automorphism must preserve C, if a property is definable then it is invariant. A cardinality argument seems to suggest there are invariant but non definable properties. But first let us see what properties are definable. Lemma 3.1.3 The property of being clopen is definable in n. Proof: The clopen sets are the complemented objects. Then P is clopen if and only if (3Q)(PvQ=1 & PAQ=O). Lemma 3.1.4 The property of being thin is definable in n. Proof: A 11H class P is thin if and only if n(P) is complemented, that is (VQ)[Q < P * (3R)(R < P & Q v R = P & Q V R = O)]. Lemma 3.1.5 the property of being a singleton is definable in n. Proof: Singletons are the atoms in n, so P is a singleton if and only if O
