Definable sets in boolean ordered o-minimal structures. II

A monadic (Boolean) algebra is a Boolean algebra A together with an operator 3 on A (called an existential quantifier, or, simply, a quantifier) such that 30=0, pfk 3p, and 3(^A 3q) = 3p* 3g whenever p and q are in A. Most of this note uses nothing more profound about monadic algebras than the definition. The reader interested in the motivation for and the basic facts in the theory of monadic algebras may, however, wish to consult [2]. Every Boolean algebra can be converted into a monadic algebra, usually in several ways. (One way is to write 3p = p for all p; another is to write Bp =0 or 1 according p =0 or py^O. These special operators are known the discrete and the simple quantifier, respectively.) It follows, a fortiori, that every Boolean algebra can be embedded into a monadic algebra, and it is clear, on grounds of universal algebra, that among the monadic extensions of a Boolean algebra there is one that is as free possible. To be more precise, let us say that a monadic algebra A is a free monadic extension of a Boolean algebra B if (i) B is a Boolean subalgebra of A, (ii) A is (monadically) generated by B, (iii) every Boolean homomorphism g that maps B into an arbitrary monadic algebra C has a (necessarily unique) extension to a monadic homomorphism f that maps A into C. The statement that a monadic extension as free possible always exists means that every Boolean algebra B has a free monadic extension A; the algebra A is uniquely determined to within a monadic isomorphism that is equal to the identity on B. The purpose of what follows is to give a constructive proof of this fact, i.e., a proof that exhibits A by means of certain set-theoretic constructions based on B (instead of a structurally not very informative proof via equivalence classes of strings of symbols). A by-product of the proof is a rather satisfying insight into the structure of the (Stone) dual space of the free monadic extension. The theorem can (and will) be formulated in such a way to subsume the results of Hyman Bass [l ] on the cardinal number of finitely generated free monadic algebras. The idea of the construction is this. Step 1: form a free Boolean algebra generated by a copy of B. Step 2: adjoin the elements of that

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle a 2-dimensional group definable in \mathcal{M} , and \mathcal{D}=\langle G;+,\ldots\rangle a strongly minimal structure, all of whose atomic relations are definable in \mathcal{M} . If \mathcal{D} is not locally modular, then an algebraically closed field K is interpretable in \mathcal{D} , and the group G , with all its induced \mathcal{D} -structure, is definably isomorphic in \mathcal{D} to an algebraic K -group with all its induced K -structure.

It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."0. Introduction.In this paper we study the structure of definable sets (of tuples) in an arbitrary O-minimal structure M (in which the underlying order is dense).Recall from [PSI, PS2] that the structure M is said to be O-minimal if M = (M, <, Ri)iei, where < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and intervals (a, b) where aE M or a = -co and b E M or b = +co.M is said to be strongly O-minimal if every N which is elementarily equivalent to M is O-minimal.We will always assume that the underlying order of M is a dense order with no first or least element.In this paper we also introduce the notion of a definable set X C Mn being definably connected, and we prove THEOREM 0.1.Let M be O-minimal.Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.THEOREM 0.2.If M is O-minimal, then M is strongly O-minimal.THEOREM 0.3.(a) Let M be O-minimal and let (p(xi,...,xn,yi,...,ym) be any formula of L (the language for M).Then there is K < uj such that for any b E Mm, the set <j>(x,b)M (= {5 E Mn:M t= 4>(a,b)}) has at most K definably connected components.(b) If M is a O-minimal expansion o/(R, <), then in (a) we can replace definably connected by connected.Theorems 0.1 and 0.2 are proved simultaneously by a rather complicated induction argument (outlined in §3 and undertaken in § §4 and 5).Theorem 0.3 follows from Theorems 0.1 and 0.2 by a compactness argument.Let us remark that if M is the field of real numbers, or more generally any real closed field, then by Tarski's quantifier elimination [T], M is (strongly) O-minimal and moreover the definable sets (of n-tuples) in M are precisely the semialgebraic

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,&lt;)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Let ℜ be an expansion of a dense linear order (R, &lt;) without endpoints having theintermediate value property, that is, for alla, b∈R, every continuous (parametrically) definable functionf: [a, b] →Rtakes on all values inRbetweenf(a) andf(b). Every expansion of the real line (ℝ, &lt;), as well as every o-minimal expansion of (R, &lt;), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, &lt;) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, &lt;, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: IfGis a proper nontrivial subgroup of (R, *) definable in ℜ, then eitherGis dense and codense inR, orGcontains an elementusuch that (R, &lt;, *,e, u, G) is elementarily equivalent to (ℚ, &lt;, +, 0, 1, ℤ), whereedenotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, &lt;, +, 0, 1, ℤ).

Let M be an o-minimal structure or a p-adically closed field. Let be the space of complete n-types over M equipped with the following topology: The basic open sets of are of the form Ũ = {p ∈ Sn (M): U ∈ p} for U an open definable subset of Mn. is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K[X1, …, Xn]; see [CR].) We will equip with a sheaf of LM-structures (where LM is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th(M) has definable Skolem functions, then if p ∈ , it follows that M(p), the definable ultrapower of M at p, can be factored through Mp, the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ Nn and f is an M-definable (partial) function defined at a, then there is an open M-definable set U ⊂ Nn with a ∈ U, and a continuous M-definable function g:U → N such that g(a) = f(a).In the case that M is an o-minimal expansion of a real closed field (or M is a p-adically closed field), it turns out that M(p) can be recovered as the unique quotient of Mp which is an elementary extension of M.

Let $K,R$ be an algebraically closed field (of characteristic zero) and a real closed field respectively with $K=R(\sqrt{-1}).$ We show that every $K$-analytic set definable in an o-minimal expansion of $R$ can be locally approximated by a sequence of $K$

The Marker-Steinhorn Theorem (cf. [2] and [3]), says the following. If T is an o-minimal theory and M ≺ N is an elementary extension of models of T such that M is Dedekind complete in N, then for every N-definable subset X of Nk, the trace X ∩ Mk is M-definable. The original proof in [2] gives an explicit method how to construct a defining formula of X ∩ Mk out of a defining formula of X. A geometric reformulation of the Marker-Steinhorn Theorem is the definability of Hausdorff limits of families of definable sets. An explicit construction of these Hausdorff limits for expansions of the real field has recently been achieved in [1]. Both proofs and also the treatment [3] are technically involved.Here we give a short algebraic, but not constructive proof, if T is an expansion of real closed fields. In fact we'll identify the statement of the Theorem with a valuation theoretic property of models of T (namely condition (†) below). Therefore our proof might be applicable to other elementary classes which expand fields, if a notion of dimension and a reasonable valuation theory are available.From now on, let T be an o-minimal expansion of real closed fields. We have to show the following (cf. [2], Th. 2.1. for this formulation). If M is a model of T and p is a tame n-type over M (i.e., M is Dedekind complete in M ⟨ᾱ⟩ := dcl(Mᾱ) for some realization ᾱ of p), then p is a definable type (cf. [4], 11 .b).

Remark on a paper of M [formula omitted]czyński

If F B(2 n − 1) denotes the Boolean algebra with 2 n − 1 free generators and P(2 n ) is the Cartesian product of 2 n Boolean algebras all equal to F B(2 n − 1), we define on P(2 n ) an existential quantifier ∃ by means of a relatively complete Boolean subalgebra of P(2 n ) and we prove that (P(2n),∃) is the monadic Boolean algebra with n free generators. Every element of P(2 n ) is a 2 n -tuple whose coordinates are in F B(2 n − 1); in particular, so are the n generators of P(2 n ). We indicate in this work the coordinates of the n generators of P(2 n ).

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion. Some of the main results of the paper are: • (1) If there is a maximal family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ. • (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products. • (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA. • (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras.

Assuming the continuum hypothesis, a set X of real numbers will be constructed such that every compact continuous image of X is countable, but XXX admits a uniformly continuous mapping onto the Cantor set. This completes (modulo the continuum hypothesis and the Ulam measure problem) the determination of whether a product or coproduct of m Boolean algebras or fields of sets can have an infinite free, or projective, or injective subor quotient object when the factors do not [1]. The concluding construction (modulo the hypothesis) is embarrassingly easy, but the result is unlike the others [1 ]; this is the only instance of a finite coproduct creating a remarkable subobject (viz. an infinite free subfield), and there is no instance among these of a finite product creating a remarkable quotient. The use made of the continuum hypothesis seems slight. However, Sierpinski has pointed out that the weaker result that there is a set of real numbers of the power of the continuum admitting no continuous mapping onto [0, 1] (C5 in [2]) has not been established without the hypothesis. On the continuum hypothesis, that is a by-product of an important construction of Lusin, the subject of Chapter II of [2]. We shall find X in the Cantor set C considered as an infinite power of the compact group Z2 (written additively). By a lemma of Lavrentiev (Theorem 99 of [3]), any continuous mapping of a subset of C onto [0, 1] can be extended over a Ga subset of C. There are only c (the power of the continuum) Ga sets, and each has only c continuous mappings to [0, 1]. Hence we can index all these mappingsfa by ordinals of smaller cardinal than c, and index similarly the points xa of C. For each fa, the inverse images of points are c disjoint subsets of C each closed in their union; so except for countably many, they are nowhere dense. By the continuum hypothesis, C is not a union of fewer than c such sets. There is no difficulty in building up sets S, T, no subset of either of which is mapped onto [0, 1] by any fa, but with every xa representable as sa+ta, Sa in S and ta in T. Carry along expanding sets Ha disjoint from S, Ka disjoint from T (Ho and Ko empty). Arrived