Elimination of Quantifiers in Algebraic Structures
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14. Least Upper Bounds in Ordered Rings Recall the Definition of a Commutative Ring with 1 from Lecture 3
14. Least upper bounds in ordered rings Recall the definition of a commutative ring with 1 from lecture 3 (Definition 3.2 ). Definition 14.1. A commutative ring R with unity consists of the following data: a set R, • two maps + : R R R (“addition”), : R R R (“multiplication”), • × −→ · × −→ two special elements 0 R = 0 , 1R = 1 R, 0 = 1 such that • ∈ 6 (1) the addition + is commutative and associative (2) the multiplication is commutative and associative (3) distributive law holds:· a (b + c) = ( a b) + ( a c) for all a, b, c R. (4) 0 is an additive identity. · · · ∈ (5) 1 is a multiplicative identity (6) for all a R there exists an additive inverse ( a) R such that a + ( a) = 0. ∈ − ∈ − Example 14.2. The integers Z, the rationals Q, the reals R, and integers modulo n Zn are all commutative rings with 1. The set M2(R) of 2 2 matrices with the usual matrix addition and multiplication is a noncommutative ring with 1 (where× “1” is the identity matrix). The set 2 Z of even integers with the usual addition and multiplication is a commutative ring without 1. Next we introduce the notion of an integral domain. In fact we have seen integral domains in lecture 3, where they were called “commutative rings without zero divisors” (see Lemma 3.4 ). Definition 14.3. A commutative ring R (with 1) is an integral domain if a b = 0 = a = 0 or b = 0 · ⇒ for all a, b R. (This is equivalent to: a b = a c and a = 0 = b = c.) ∈ · · 6 ⇒ Example 14.4. -
A Note on Embedding a Partially Ordered Ring in a Division Algebra William H
proceedings of the american mathematical society Volume 37, Number 1, January 1973 A NOTE ON EMBEDDING A PARTIALLY ORDERED RING IN A DIVISION ALGEBRA WILLIAM H. REYNOLDS Abstract. If H is a maximal cone of a ring A such that the subring generated by H is a commutative integral domain that satisfies a certain centrality condition in A, then there exist a maxi- mal cone H' in a division ring A' and an order preserving mono- morphism of A into A', where the subring of A' generated by H' is a subfield over which A' is algebraic. Hypotheses are strengthened so that the main theorems of the author's earlier paper hold for maximal cones. The terminology of the author's earlier paper [3] will be used. For a subsemiring H of a ring A, we write H—H for {x—y:x,y e H); this is the subring of A generated by H. We say that H is a u-hemiring of A if H is maximal in the class of all subsemirings of A that do not contain a given element u in the center of A. We call H left central if for every a e A and he H there exists h' e H with ah=h'a. Recall that H is a cone if Hn(—H)={0], and a maximal cone if H is not properly contained in another cone. First note that in [3, Theorem 1] the commutativity of the hemiring, established at the beginning of the proof, was only exploited near the end of the proof and was not used to simplify the earlier details. -
Algorithmic Semi-Algebraic Geometry and Topology – Recent Progress and Open Problems
Contemporary Mathematics Algorithmic Semi-algebraic Geometry and Topology – Recent Progress and Open Problems Saugata Basu Abstract. We give a survey of algorithms for computing topological invari- ants of semi-algebraic sets with special emphasis on the more recent devel- opments in designing algorithms for computing the Betti numbers of semi- algebraic sets. Aside from describing these results, we discuss briefly the back- ground as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems. Contents 1. Introduction 1 2. Semi-algebraic Geometry: Background 3 3. Recent Algorithmic Results 10 4. Algorithmic Preliminaries 12 5. Topological Preliminaries 22 6. Algorithms for Computing the First Few Betti Numbers 41 7. The Quadratic Case 53 8. Betti Numbers of Arrangements 66 9. Open Problems 69 Acknowledgment 71 References 71 1. Introduction This article has several goals. The primary goal is to provide the reader with a thorough survey of the current state of knowledge on efficient algorithms for computing topological invariants of semi-algebraic sets – and in particular their Key words and phrases. Semi-algebraic Sets, Betti Numbers, Arrangements, Algorithms, Complexity . The author was supported in part by NSF grant CCF-0634907. Part of this work was done while the author was visiting the Institute of Mathematics and its Applications, Minneapolis. 2000 Mathematics Subject Classification Primary 14P10, 14P25; Secondary 68W30 c 0000 (copyright holder) 1 2 SAUGATA BASU Betti numbers. At the same time we want to provide graduate students who intend to pursue research in the area of algorithmic semi-algebraic geometry, a primer on the main technical tools used in the recent developments in this area, so that they can start to use these themselves in their own work. -
Embedding Two Ordered Rings in One Ordered Ring. Part I1
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 2, 341-364 (1966) Embedding Two Ordered Rings in One Ordered Ring. Part I1 J. R. ISBELL Department of Mathematics, Case Institute of Technology, Cleveland, Ohio Communicated by R. H. Bruck Received July 8, 1965 INTRODUCTION Two (totally) ordered rings will be called compatible if there exists an ordered ring in which both of them can be embedded. This paper concerns characterizations of the ordered rings compatible with a given ordered division ring K. Compatibility with K is equivalent to compatibility with the center of K; the proof of this depends heavily on Neumann’s theorem [S] that every ordered division ring is compatible with the real numbers. Moreover, only the subfield K, of real algebraic numbers in the center of K matters. For each Ks , the compatible ordered rings are characterized by a set of elementary conditions which can be expressed as polynomial identities in terms of ring and lattice operations. A finite set of identities, or even iden- tities in a finite number of variables, do not suffice, even in the commutative case. Explicit rules will be given for writing out identities which are necessary and sufficient for a commutative ordered ring E to be compatible with a given K (i.e., with K,). Probably the methods of this paper suffice for deriving corresponding rules for noncommutative E, but only the case K,, = Q is done here. If E is compatible with K,, , then E is embeddable in an ordered algebra over Ks and (obviously) E is compatible with the rational field Q. -
Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields
Model Theory, Algebra, and Geometry MSRI Publications Volume 39, 2000 Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields JAN DENEF Abstract. We survey applications of quantifier elimination to number the- ory and algebraic geometry, focusing on results of the last 15 years. We start with the applications of p-adic quantifier elimination to p-adic inte- gration and the rationality of several Poincar series related to congruences f(x) = 0 modulo a prime power, where f is a polynomial in several vari- ables. We emphasize the importance of p-adic cell decomposition, not only to avoid resolution of singularities, but especially to obtain much stronger arithmetical results. We survey the theory of p-adic subanalytic sets, which is needed when f is a power series instead of a polynomial. Next we explain the fundamental results of Lipshitz{Robinson and Gardener{Schoutens on subanalytic sets over algebraically closed complete valued fields, and the connection with rigid analytic geometry. Finally we discuss recent geo- metric applications of quantifier elimination over C((t)), related to the arc space of an algebraic variety. One of the most striking applications of the model theory of valued fields to arithmetic is the work of Ax and Kochen [1965a; 1965b; 1966; Kochen 1975], and of Ershov [1965; 1966; 1967], which provided for example the first quantifier elimination results for discrete valued fields [Ax and Kochen 1966], and the decidability of the field Qp of p-adic numbers. As a corollary of their work, Ax and Kochen [1965a] proved the following celebrated result: For each prime number p, big enough with respect to d, any homogeneous polynomial of degree 2 d over Qp in d + 1 variables has a nontrivial zero in Qp. -
More Model Theory Notes Miscellaneous Information, Loosely Organized
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants. -
Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 4, October 1996 MODEL COMPLETENESS RESULTS FOR EXPANSIONS OF THE ORDERED FIELD OF REAL NUMBERS BY RESTRICTED PFAFFIAN FUNCTIONS AND THE EXPONENTIAL FUNCTION A. J. WILKIE 1. Introduction Recall that a subset of Rn is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form α~ Rn : p(α~)=0, { ∈ } α~ Rn:q(α~)>0 where p(~x), q(~x)aren-variable polynomials with real co- { ∈ } efficients. A map from Rn to Rm is called semi-algebraic if its graph, considered as a subset of Rn+m, is so. The geometry of such sets and maps (“semi-algebraic geometry”) is now a widely studied and flourishing subject that owes much to the foundational work in the 1930s of the logician Alfred Tarski. He proved ([11]) that the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic. (A familiar simple instance: the image of a, b, c, x R4 :a=0andax2 +bx+c =0 {h i∈ 6 } under the projection map R3 R R3 is a, b, c R3 :a=0andb2 4ac 0 .) Tarski’s result implies that the× class→ of semi-algebraic{h i∈ sets6 is closed− under≥ first-} order logical definability (where, as well as boolean operations, the quantifiers “ x R ...”and“x R...” are allowed) and for this reason it is known to ∃ ∈ ∀ ∈ logicians as “quantifier elimination for the ordered ring structure on R”. Immedi- ate consequences are the facts that the closure, interior and boundary of a semi- algebraic set are semi-algebraic. -
Arxiv:2007.09244V1 [Math.LO] 17 Jul 2020 Rdcs Given Products
AXIOMS FOR COMMUTATIVE UNITAL RINGS ELEMENTARILY EQUIVALENT TO RESTRICTED PRODUCTS OF CONNECTED RINGS JAMSHID DERAKHSHAN AND ANGUS MACINTYRE† Abstract. We give axioms in the language of rings augmented by a 1-ary predicate symbol Fin(x) with intended interpretation in the Boolean algebra of idempotents as the ideal of finite elements, i.e. finite unions of atoms. We prove that any commutative unital ring satisfying these axioms is elementarily equivalent to a restricted product of connected rings. This is an extension of the results in [3] for products. While the results in [3] give a converse to the Feferman-Vaught theorem for products, our results prove the same for restricted products. We give a complete set of axioms in the language of rings for the ring of adeles of a number field, uniformly in the number field. 0. Introduction This paper is a natural sequel to [3] and the main results and proofs are natural extensions of those in [3]. In many cases we will simply refer to the material from [3] [3] deals with the model theory of products of connected unital rings, and can be construed as providing a partial converse to the Feferman-Vaught Theorem [9] in the special case of products Qi∈I Ri, i ∈ I, where Ri are connected commutative unital rings and I an index set (Recall that a commutative ring R is connected if 0, 1 are the only idempotents of R). The converse concerns the issue of providing axioms for rings elementarily equivalent to rings Qi∈I Ri as above. The solution of this problem is given in [3] and, inter alia has applications to non-standard models arXiv:2007.09244v1 [math.LO] 17 Jul 2020 of PA (first order Peano arithmetic) in [4]. -
With Squares Positive and an /-Superunit Can Be Embedded in a Unital /-Prime Lattice-Ordered Ring with Squares Positive
PROCEEDINGSof the AMERICANMATHEMATICAL SOCIETY Volume 121, Number 4, August 1994 THE UNITABILITY OF /-PRIME LATTICE-ORDEREDRINGS WITH SQUARESPOSITIVE MA JINGJING (Communicated by Lance W. Small) Abstract. It is shown that an /-prime lattice-ordered ring with squares positive and an /-superunit can be embedded in a unital /-prime lattice-ordered ring with squares positive. In [1, p. 325] Steinberg asked if an /-prime /-ring with squares positive and an /-superunit can be embedded in a unital /-prime /-ring with squares positive. In this paper we show that the answer is yes. If R is a lattice-ordered ring (/-ring) and a e R+, then a is called an /- element of R if b A c = 0 implies ab/\c = baAc = 0. Let T = T(R) = {a e R : \a\ is an /-element of R}. Then T is a convex /-subring of R, and R is a subdirect product of totally ordered T-T bimodules [2, Lemma 1]. R is an /-ring precisely when T = R. Throughout this paper T will denote the subring of /-elements of R. An element e > 0 of an /-ring R is called a superunit if ex > x and xe > x for each x e R+ ; e is an /-superunit if it is a superunit and an /-element. R is infinitesimal if x2 < x for each x in R+ . The /-ring R is called /-prime if the product of two nonzero /-ideals is nonzero and /-semiprime if it has no nonzero nilpotent /-ideals. An /-ring R is called squares positive if a2 > 0 for each a in R. -
Convex Extensions of Partially Ordered Rings
Convex Extensions of Partially Ordered Rings Niels Schwartz*) A series of lectures given at the conference "Géométrie algébrique et analytique réelle" Kenitra, Morocco, September 13 – 20, 2004 This series of lectures is about algebraic aspects of real algebraic geometry. Partially ordered rings (porings) occur very naturally in real algebraic geometry, as well as in other branches of mathematics. There is a rich classical theory of porings (the beginnings dating back to the 1930s and 1940s). To a large extent it was inspired by the study of rings of continuous functions in analysis and topology. I will mention rings of continuous functions from time to time, but will not attempt an in-depth discussion. With the rise of real algebraic geometry in the last 25 years there also arose a need for algebraic methods that are specifically adapted to real algebraic geometry, as opposed to general algebraic geometry. For a long time commutative algebra has been the answer to a similar need in general algebraic geometry. For real algebraic geometry, an important part of the answer is the theory of porings. However, the classical theory of porings frequently does not address the questions from real algebraic geometry. Therefore it is necessary to develop a more comprehensive theory of porings that includes rings of continuous functions, but also deals with phenomena occurring in real geometry. This is a major task that is not nearly finished at the present time. First I give the definition of porings: Definition 1 A partially ordered ring (poring) is a ring A (all rings are commutative with 1) together with a subset P ! A that satisfies the following conditions • P + P ! P ; • P ! P " P ; • A2 ! P ; • P ! "P = {0} . -
Finitely Presented and Coherent Ordered Modules and Rings Friedrich Wehrung
Finitely presented and coherent ordered modules and rings Friedrich Wehrung To cite this version: Friedrich Wehrung. Finitely presented and coherent ordered modules and rings. Communications in Algebra, Taylor & Francis, 1999, 27 (12), pp.5893–5919. 10.1080/00927879908826797. hal-00004049 HAL Id: hal-00004049 https://hal.archives-ouvertes.fr/hal-00004049 Submitted on 24 Jan 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FINITELY PRESENTED AND COHERENT ORDERED MODULES AND RINGS F. WEHRUNG Abstract. We extend the usual definition of coherence, for mod- ules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features: (i) Every subring of Q, and every totally ordered division ring, is po-coherent. (ii) For a partially ordered right module A over a po-coherent po- ring R, A is po-coherent if and only if A is a finitely presented R-module and A+ is a finitely generated R+-semimodule. -
Quantifier Elimination and Real Closed Ordered Fields with A
Quanti¯er Elimination and Real Closed Ordered Fields with a Predicate for the Powers of Two Yimu Yin December 17, 2005 Abstract In this thesis we ¯rst review the model theory of quanti¯er elim- ination and investigate the logical relations among various quanti¯er elimination tests. In particular we prove the equivalence of two quanti- ¯er elimination tests for countable theories. Next we give a procedure for eliminating quanti¯ers for the theory of real closed ordered ¯elds with a predicate for the powers of two. This result was ¯rst obtained by van den Dries [20]. His method is model-theoretic, which provides no apparent bounds on the complexity of a decision procedure. In the last section we give a complete axiomatization of the theory of real closed ordered ¯elds with a predicate for the Fibonacci numbers. 1 Acknowledgements I thank my advisor Jeremy Avigad for the guidance he has provided me. I am grateful to James Cummings and Rami Grossberg for many helpful suggestions. I also thank Chris Miller for suggesting the problem considered in the last section of this thesis. 2 1 Introduction A \decision procedure" for mathematics is, roughly, a procedure for deter- mining whether or not a mathematical statement is true. Of course, any proof can be viewed as a procedure for determining that a particular theo- rem is true; but when we talk about decision procedures, we usually mean a procedure for deciding a class of statements. Some decision procedures have been around for a very long time. For example, Euclid's algorithm is such a procedure.