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Heirs of box types in polynomially bounded structures; preprint, submitted for publication
"... 2. Heirs of cuts in polynomially bounded structures 3. Box types 4. The box type associated to a cut ..."
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2. Heirs of cuts in polynomially bounded structures 3. Box types 4. The box type associated to a cut
STABILITY OF THE THEORY OF EXISTENTIALLY CLOSED SACTS OVER A RIGHT COHERENT MONOID S
"... For Professor L.A. Bokut on the occasion of his 70th birthday Abstract. Let LS denote the language of (right) Sacts over a monoid S and let ΣS be a set of sentences in LS which axiomatises Sacts. A general result of model theory says that ΣS has a model companion, denoted by TS, precisely when the ..."
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For Professor L.A. Bokut on the occasion of his 70th birthday Abstract. Let LS denote the language of (right) Sacts over a monoid S and let ΣS be a set of sentences in LS which axiomatises Sacts. A general result of model theory says that ΣS has a model companion, denoted by TS, precisely when the class E of existentially closed Sacts is axiomatisable and in this case, TS axiomatises E. It is known that TS exists if and only if S is right coherent. Moreover, by a result of Ivanov, TS has the modeltheoretic property of being stable. In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over TS algebraically, thus reducing our examination of TS to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that TS is stable and to prove another result of Ivanov, namely that TS is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that TS is totally transcendental and is such that the Urank of any type coincides with its Morley rank. 1.
Forking and Multiplicity in First Order Theories
, 2001
"... Our style will be to write definitions and theorems. In most cases, we will only sketch the proofs and refer the reader to exactly where the proof can be found usually in [1, 3, 9]. ..."
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Our style will be to write definitions and theorems. In most cases, we will only sketch the proofs and refer the reader to exactly where the proof can be found usually in [1, 3, 9].
DEFINABLE STRUCTURES IN OMINIMAL THEORIES: ONE DIMENSIONAL TYPES
"... Abstract. Let N be a structure definable in an ominimal structureM and p ∈ SN (N), a complete N1type. If dimM(p) = 1 then p supports a combinatorial pregeometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is nonorthogonal to a generic type in an N ..."
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Abstract. Let N be a structure definable in an ominimal structureM and p ∈ SN (N), a complete N1type. If dimM(p) = 1 then p supports a combinatorial pregeometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is nonorthogonal to a generic type in an Ndefinable (possibly ordered) group whose structure is linear, or, if p is rich then p is nonorthogonal to a generic type of an Ndefinable real closed field. As a result we obtain a similar trichotomy for definable onedimensional structures in ominimal theories. In this paper we prove a trichotomy theorem for onedimensional types in structures definable in ominimal theories. With this we conclude the work started in [4], of which this is a direct continuation. Recall that a structure N is said to be definable in an ominimal structure M if the universe N, of N, as well as all its atomic relations, are definable sets (possibly of several variables, possibly using parameters) in the structure M. In [4] we proved a weak version of Zilber’s trichotomy: Theorem 1. Let N be a stable structure definable in an ominimal structure M. If dimM(N) = 1 then N is 1based. The local nature of phenomena in ominimal theories does not leave room for more precise global statements in the unstable case. The aim of this paper is to remedy this situation by applying the results of [4] and [2] to obtain a complete classification of 1Mdimensional types in N without any additional global assumptions on N. Our main result can be summed up by (see definitions below): Theorem 2. Let M be an ominimal structure and N definable in M. Let p ∈ SN (N) be oneMdimensional. Then exactly one of the following holds: (1) p is trivial. (2) p is linear, in which case p is nonorthogonal to a generic type of an Ndefinable (possibly locally ordered) group G. The structure which N induces on G is linear, i.e., given by definable (possibly local) subgroups of Gn. (3) p is rich, in which case it is nonorthogonal to a generic type of an Ndefinable real closed field R. In fact, our results will be more precise and give a stable/unstable dichotomy (see Theorem 2.1). As a corollary to the above we can complete the analysis of definable one dimensional structures which began in [4]:
And by contacting: The MIMS Secretary
, 2008
"... Heirs of box types in polynomially bounded structures ..."
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Reducts and Expansions of Stable and Simple Theories
, 2004
"... c ○ This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author’s prior consent. In this thesis ..."
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c ○ This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author’s prior consent. In this thesis we will study certain properties like simplicity, categoricity or CMtriviality of reducts and Skolem expansions of simple and stable theories. The first part is about Skolem expansions of simple theories. We will show that if we add the Skolem function for an algebraic formula to an algebraically bounded, modelcomplete, simple theory T, then its modelcompletion T ∗, which we know exists from Winkler’s Theorem, is again simple. If T is ωcategorical then so is T ∗. This will give us a method to turn algebraic closure into definable closure without losing simplicity or ωcategoricity. We illustrate with an example that if T is uncountably categorical then T ∗ need not to be. After that we examine the case of adding the Skolem function for a non algebraic
unknown title
, 1989
"... In this paper, automorphism groups are studied, and variations on the small index property are proved for several kinds of structures. The main results are: (1) it is consistent with ZF to assume that every countable structure has the small index property; (2) (assuming the continuum hypothesis) if ..."
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In this paper, automorphism groups are studied, and variations on the small index property are proved for several kinds of structures. The main results are: (1) it is consistent with ZF to assume that every countable structure has the small index property; (2) (assuming the continuum hypothesis) if 3K is a saturated Stable structure of cardinality K, and if H is a normal subgroup of the group Aut(9W) of automorphisms of 3K whose index is at most K,, then H contains every strong automorphism of 2ft; (3) (with the continuum hypothesis) if 2TJ is an (ostable saturated structure of cardinality Kt and if H is a subgroup of Aut(3ft) of index at most Kt, then there exists a countable subset A of 2R such that every automorphism of 3ft leaving A pointwise fixed is in H; (4) the same thing is true for the dense linear ordering. 1.