Coordinatization in Superstable Theories. I. Stationary Types

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \cdots \times S_m \subset \mathbb{C}^n$ be the multigrid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of the combinatorial nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo--Lipton--Schwartz--Zippel lemma holds, except for a special family of polynomials that we call $\lambda$-reducible. This yields a simultaneous generalization of the Szemerédi--Trotter theorem and the Schwartz--Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $\lambda$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a Cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $\lambda$-reducible polynomials, which we leave as an open problem.

In this final chapter we give some scattered results on superstable theories. One of the main lines here is the development of some technology which helps towards generalizing results from the finite rank context to the general superstable context. A key point is the interaction between the local theory developed in Chapter 7 and Shelah’s ‘infinity rank’, R∞. This will yield in some situations ‘continuity and definability’ of p-weight. In any case one of the goals of this chapter is to suitably generalize Buechler’s dichotomy theorem to regular types in superstable theories. This is done in section 5, under the assumption that T is countable and has NDOP and NOTOP (two structure properties); this result has great importance for some fine questions concerning the spectrum function. Note we also give some other versions of the dichotomy theorem, which do not depend on assuming NDOP and NOTOP, and which are relevant to Vaught’s conjecture for superstable theories. In the process of developing the technology, we mention various related ‘non-geometric’ results, such as the existence of regular types between arbitrary models, and definability questions in non-multidimensional theories.

1: Structures. 1.1. Structures. 1.2. Sentences. 1.3. Embeddings. 1.4. The Compactness Theorem. 1.5. Elementary classes and theories. 1.6. Complete theories. 1.7. Definable sets. 1.8. References. 2: Quantifier Elimination. 2.1. Elimination sets. 2.2. Discrete linear orders. 2.3. Dense linear orders. 2.4. Algebraically closed fields (and Tarski). 2.5. Tarski again: Real closed fields. 2.6. pp-elimination of quantifiers and modules. 2.7. Strongly minimal theories. 2.8. o-minimal theories. 2.9. Computational aspects of q. e. 2.10. References. 3: Model Completeness. 3.1. An introduction. 3.2. Abraham Robinson's test. 3.3. Model completeness and algebra. 3.4. p-adic fields and Artin's conjecture. 3.5. Existentially closed fields. 3.6. DCF0. 3.7. SCFp and DCFp. 3.8. ACFA. 3.9. References. 4: Elimination of Imaginaries. 4.1. Interpretability. 4.2. Imaginary elements. 4.3. Algebraically closed fields. 4.4. Real closed fields. 4.5. The elimination of imaginaries sometimes fails. 4.6. References. 5: Morley Rank. 5.1. A tale of two chapters. 5.2. Definable sets. 5.3. Types. 5.4. Saturated models. 5.5. A parenthesis: pure injective models. 5.6. Omitting types. 5.7. The Morley rank, at last. 5.8. Strongly minimal sets. 5.9. Algebraic closure and definable closure. 5.10. References. 6: Omega-stability. 6.1. Totally transcendental theories. 6.2. omega-stable groups. 6.3. omega-stable fields. 6.4. Prime models. 6.5. DCF0 revisited. 6.6. Ryll-Nardzewski's Theorem and other things. 6.7. References. 7: Classifying. 7.1. Shelah's Classification Theory. 7.2. Simple theories. 7.3. Stable theories. 7.4. Superstable theories. 7.5. w-stable theories. 7.6. Classifiable theories. 7.7. Shelah's Uniqueness Theorem. 7.8. Morley's Theorem. 7.9. Biinterpretability and Zilber Conjecture. 7.10. Two algebraic examples. 7.11. References. 8: Model Theory and Algebraic Geometry. 8.1. Introduction. 8.2. Algebraic varieties, ideals, types. 8.3. Dimension and Morley rank. 8.4. Morphisms and definable functions. 8.5. Manifolds. 8.6. Algebraic groups. 8.7. The Mordell-Lang Conjecture. 8.8. References. 9: O-Minimality. 9.1. Introduction. 9.2. The Monotonicity Theorem. 9.3. Cells. 9.4. Cell decomposition and other theorems. 9.5. Their proofs. 9.6. Definable groups in o-minimal structures. 9.7. O-minimality and Real Analysis. 9.8. Variants on the o-minimal theme. 9.9. No rose without thorns. 9.10. References. Bibliography. Index.

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations ρ1,⋯,ρk, is the downward closed set Av(ρ1,⋯,ρk) consisting of all equivalence relations which do not contain any of ρ1,⋯,ρk: (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?

Computer programing for construction and enumeration of all regular topologies and equivalence relations on finite sets

LOGICAL REASONING. Symbolic Language. Two Quantifiers. Five Logical Operators. Laws of Logic. Logic Circuits. Translations. WRITING OUR REASONING. Proofs and Arguments. Proving Implications. Writing a Proof. Working with Quantifiers. Using Cases. Proof by Contradiction. Mathematical Induction. Axiomatic Systems. SETS - THE BUILDING BLOCKS. Sets and Elements. Operations on Sets. Multiple Unions and Intersections. Cross Product. Finite Sets. Infinite Sets. RELATIONS - THE ACTION. Relations. Equivalence Relations. Functions. Order Relations. Summary. Appendices. Bibliography.

The topics in this paper deal with generating subsemigroups of the full transformation semigroup????(????) on a totally ordered finite set (????, ≤) with an arbitrary equivalence relation ???? defined on it.Using the equivalence relation on the set and the cardinality of the set, we investigate under whatconditions some of these subsemigroups of ????(????) generated are equal.

We investigate definable automorphisms of ^(uj/fin and show that e.g. every Borel automorphism is trivial. The existence of nontrivial projective automor- phisms is consistent and independent from ZFC + CH. Throughout this paper a2(w)/fin will denote the Boolean algebra of 0>(to) modulo the ideal of finite sets. Call an automorphism F of ^(c^/fin trivial if there are two cofinite sets a,b e u and a bijection /: a -> b such that F(x) = (f(a n x)). Here (x) denotes the equivalence class of x under the equivalence relation x =+ y iff (x\y)U (y\x) is finite. Note that there are exactly 2K° trivial automorphisms. It is a well-known fact, first proved by W. Rudin (7), that the Continuum Hypothesis implies there are 22 ° automorphisms of £P(o:)/nn, hence most of them are nontrivial. It is also consistent with —,CH that there is a nontrivial automorphism of aa(w)/fin. This was proved by Baumgartner (unpublished). He noticed that if there is a family {aa: a < ux) of subsets of w such that (i) aa as is finite for every a < s < «,, (ii) {aa: a<to1}U(to)< generates a maximal ideal /, then one can build by induction a family {fa: a < tcx} of functions such that (iii) fa: aa -* aa is a permutation without fixed points,

A choice space is a finite set of alternatives endowed with a map associating to each menu a nonempty subset of selected items. A congruence on a choice space is an equivalence relation that preserves its structure. Intuitively, two alternatives are congruent if the agent is indifferent between them, and, in addition, her choice is influenced by them in exactly the same way. We give an axiomatic characterization of the notion of congruence in terms of three natural conditions: binary fungibility, common destiny, and repetition irrelevance. Further, we show that any congruence satisfies the following desirable properties: (hereditariness) it induces a well-defined choice on the quotient set of equivalence classes; (reflectivity) the primitive behavior can be always retrieved from the quotient choice, regardless of any feature of rationality; (consistency) all basic axioms of choice consistency are preserved back and forth by passing to the quotient. We also prove that the family of all congruences on a choice space forms a lattice under set-inclusion, having equality as a minimum, and a unique maximum, called revealed indiscernibility. The latter relation can be seen as a limit form of revealed similarity as the agent’s rationality increases.

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\in K[X]$ such that $f(\Omega)\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring ${\rm Int}_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose image over $S$ is contained in $A$. We show that the integral closure of ${\rm Int}_K(S,A)$ is equal to the contraction to $K[X]$ of ${\rm Int}(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elements over $D$ contained in an algebraic closure $\bar K$ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of ${\rm Int}_K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form $D[X]+p(X)K[X]$, where $p(X)$ is a monic non-constant polynomial over $D$: we prove that the integral closure of such a pullback is equal to the ring of polynomials over $K$ which are integral-valued over the set of roots $\Omega_p$ of $p(X)$ in $\bar K$.

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $L^1$ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor. The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon--Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

Set Labelling Vertices To Ensure Adjacency Coincides With Disjointness

The problem of assessing the similarity of two groupings of data into categories is considered. Such groupings may arise as a consequence of subjective categorisation by different people or by repeated machine classification by an algorithm which is dependent on the order of data submission or even non-deterministic. Several self organising classification algorithms (e.g. fuzzy ART and fuzzy Min-Max) suffer from this property. In either case the question of the reliability of the categorisation can only be answered if some notion of distance between different categorisations is available. (This is because the question of reliability of the method does not depend on always producing exactly the same classification but merely on producing sufficiently close classifications). With such a notion it is possible to conduct experimental work on any data set to ensure that there is no important variation in the categorisation depending on the order in which observations are submitted to the classifier. Several different methods of measuring distance are proposed and investigated. Importantly two of these measures are conducted in the appropriate algebraic setting in which to compare partitions of sets-the lattice of equivalence relations on a finite set. Whilst this involves some computational difficulties it is hoped that the benefits of an appropriate measure outweigh them. A comparison of two of these distance measures using a known data set from the literature is performed, and the results reported.

On representing Mn's by congruence lattice of finite algebras

A signalizer functor theorem for groups of finite Morley rank