On the Theory of Left/Right Almost Groups and Hypergroups with Their Relevant Enumerations
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
Mathematics 144 Set Theory Fall 2012 Version
MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction -
Biography Paper – Georg Cantor
Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies. -
Primitive Near-Rings by William
PRIMITIVE NEAR-RINGS BY WILLIAM MICHAEL LLOYD HOLCOMBE -z/ Thesis presented to the University of Leeds _A tor the degree of Doctor of Philosophy. April 1970 ACKNOWLEDGEMENT I should like to thank Dr. E. W. Wallace (Leeds) for all his help and encouragement during the preparation of this work. CONTENTS Page INTRODUCTION 1 CHAPTER.1 Basic Concepts of Near-rings 51. Definitions of a Near-ring. Examples k §2. The right modules with respect to a near-ring, homomorphisms and ideals. 5 §3. Special types of near-rings and modules 9 CHAPTER 2 Radicals and Semi-simplicity 12 §1. The Jacobson Radicals of a near-ring; 12 §2. Basic properties of the radicals. Another radical object. 13 §3. Near-rings with descending chain conditions. 16 §4. Identity elements in near-rings with zero radicals. 20 §5. The radicals of related near-rings. 25. CHAPTER 3 2-primitive near-rings with identity and descending chain condition on right ideals 29 §1. A Density Theorem for 2-primitive near-rings with identity and d. c. c. on right ideals 29 §2. The consequences of the Density Theorem 40 §3. The connection with simple near-rings 4+6 §4. The decomposition of a near-ring N1 with J2(N) _ (0), and d. c. c. on right ideals. 49 §5. The centre of a near-ring with d. c. c. on right ideals 52 §6. When there are two N-modules of type 2, isomorphic in a 2-primitive near-ring? 55 CHAPTER 4 0-primitive near-rings with identity and d. c. -
On One-Sided Prime Ideals
Pacific Journal of Mathematics ON ONE-SIDED PRIME IDEALS FRIEDHELM HANSEN Vol. 58, No. 1 March 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 58, No. 1, 1975 ON ONE-SIDED PRIME IDEALS F. HANSEN This paper contains some results on prime right ideals in weakly regular rings, especially V-rings, and in rings with restricted minimum condition. Theorem 1 gives information about the structure of V-rings: A V-ring with maximum condition for annihilating left ideals is a finite direct sum of simple V-rings. A characterization of rings with restricted minimum condition is given in Theorem 2: A nonprimitive right Noetherian ring satisfies the restricted minimum condition iff every critical prime right ideal ^(0) is maximal. The proof depends on the simple observation that in a nonprimitive ring with restricted minimum condition all prime right ideals /(0) contain a (two-sided) prime ideal ^(0). An example shows that Theorem 2 is not valid for right Noetherian primitive rings. The same observation on nonprimitive rings leads to a sufficient condition for rings with restricted minimum condition to be right Noetherian. It remains an open problem whether there exist nonnoetherian rings with restricted minimum condition (clearly in the commutative case they are Noetherian). Theorem 1 is a generalization of the well known: A right Goldie V-ring is a finite direct sum of simple V-rings (e.g., [2], p. 357). Theorem 2 is a noncommutative version of a result due to Cohen [1, p. 29]. Ornstein has established a weak form in the noncommuta- tive case [11, p. 1145]. In §§1,2,3 the unity in rings is assumed (except in Proposition 2.1), but most of the results are valid for rings without unity, as shown in §4. -
SRB MEASURES for ALMOST AXIOM a DIFFEOMORPHISMS José Alves, Renaud Leplaideur
SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS José Alves, Renaud Leplaideur To cite this version: José Alves, Renaud Leplaideur. SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS. 2013. hal-00778407 HAL Id: hal-00778407 https://hal.archives-ouvertes.fr/hal-00778407 Preprint submitted on 20 Jan 2013 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. SRB MEASURES FOR ALMOST AXIOM A DIFFEOMORPHISMS JOSE´ F. ALVES AND RENAUD LEPLAIDEUR Abstract. We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are \long enough", then f admits a probability SRB measure. Contents 1. Introduction 2 1.1. Background 2 1.2. Statement of results 2 1.3. Overview 4 2. Markov rectangles 5 2.1. Neighborhood of critical zone 5 2.2. First generation of rectangles 6 2.3. Second generation rectangles 7 2.4. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Bounding, Splitting and Almost Disjointness
Bounding, splitting and almost disjointness Jonathan Schilhan Advisor: Vera Fischer Kurt G¨odelResearch Center Fakult¨atf¨urMathematik, Universit¨atWien Abstract This bachelor thesis provides an introduction to set theory, cardinal character- istics of the continuum and the generalized cardinal characteristics on arbitrary regular cardinals. It covers the well known ZFC relations of the bounding, the dominating, the almost disjointness and the splitting number as well as some more recent results about their generalizations to regular uncountable cardinals. We also present an overview on the currently known consistency results and formulate open questions. 1 CONTENTS Contents Introduction 3 1 Prerequisites 5 1.1 Model Theory/Predicate Logic . .5 1.2 Ordinals/Cardinals . .6 1.2.1 Ordinals . .7 1.2.2 Cardinals . .9 1.2.3 Regular Cardinals . 10 2 Cardinal Characteristics of the continuum 11 2.1 The bounding and the dominating number . 11 2.2 Splitting and mad families . 13 3 Cardinal Characteristics on regular uncountable cardinals 16 3.1 Generalized bounding, splitting and almost disjointness . 16 3.2 An unexpected inequality between the bounding and splitting numbers . 18 3.2.1 Elementary Submodels . 18 3.2.2 Filters and Ideals . 19 3.2.3 Proof of Theorem 3.2.1 (s(κ) ≤ b(κ)) . 20 3.3 A relation between generalized dominating and mad families . 23 References 28 2 Introduction @0 Since it was proven that the continuum hypothesis (2 = @1) is independent of the axioms of ZFC, the natural question arises what value 2@0 could take and it was shown @0 that 2 could be consistently nearly anything. -
Almost-Free E-Rings of Cardinality ℵ1 Rudige¨ R Gob¨ El, Saharon Shelah and Lutz Strung¨ Mann
Sh:785 Canad. J. Math. Vol. 55 (4), 2003 pp. 750–765 Almost-Free E-Rings of Cardinality @1 Rudige¨ r Gob¨ el, Saharon Shelah and Lutz Strung¨ mann Abstract. An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R+ is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater @ than 2 0 is undecidable in ZFC. While they exist in Godel'¨ s universe, they do not exist in other models @ of set theory. For a regular cardinal @1 ≤ λ ≤ 2 0 we construct E-rings of cardinality λ in ZFC which have @1-free additive structure. For λ = @1 we therefore obtain the existence of almost-free E-rings of cardinality @1 in ZFC. 1 Introduction + Recall that a unital ring R is an E-ring if the evaluation map ": EndZ(R ) ! R given by ' 7! '(1) is a bijection. Thus every endomorphism of the abelian group R+ is multiplication by some element r 2 R. E-rings were introduced by Schultz [20] and easy examples are subrings of the rationals Q or pure subrings of the ring of p-adic integers. Schultz characterized E-rings of finite rank and the books by Feigelstock [9, 10] and an article [18] survey the results obtained in the eighties, see also [8, 19]. In a natural way the notion of E-rings extends to modules by calling a left R-module M an E(R)-module or just E-module if HomZ(R; M) = HomR(R; M) holds, see [1]. -
The Axiom of Choice and Its Implications
THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial. -
Binary Operations
Binary operations 1 Binary operations The essence of algebra is to combine two things and get a third. We make this into a definition: Definition 1.1. Let X be a set. A binary operation on X is a function F : X × X ! X. However, we don't write the value of the function on a pair (a; b) as F (a; b), but rather use some intermediate symbol to denote this value, such as a + b or a · b, often simply abbreviated as ab, or a ◦ b. For the moment, we will often use a ∗ b to denote an arbitrary binary operation. Definition 1.2. A binary structure (X; ∗) is a pair consisting of a set X and a binary operation on X. Example 1.3. The examples are almost too numerous to mention. For example, using +, we have (N; +), (Z; +), (Q; +), (R; +), (C; +), as well as n vector space and matrix examples such as (R ; +) or (Mn;m(R); +). Using n subtraction, we have (Z; −), (Q; −), (R; −), (C; −), (R ; −), (Mn;m(R); −), but not (N; −). For multiplication, we have (N; ·), (Z; ·), (Q; ·), (R; ·), (C; ·). If we define ∗ ∗ ∗ Q = fa 2 Q : a 6= 0g, R = fa 2 R : a 6= 0g, C = fa 2 C : a 6= 0g, ∗ ∗ ∗ then (Q ; ·), (R ; ·), (C ; ·) are also binary structures. But, for example, ∗ (Q ; +) is not a binary structure. Likewise, (U(1); ·) and (µn; ·) are binary structures. In addition there are matrix examples: (Mn(R); ·), (GLn(R); ·), (SLn(R); ·), (On; ·), (SOn; ·). Next, there are function composition examples: for a set X,(XX ; ◦) and (SX ; ◦). -
Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected]
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Research Repository @ WVU (West Virginia University) Faculty Scholarship 1995 Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/faculty_publications Part of the Mathematics Commons Digital Commons Citation Ciesielski, Krzysztof, "Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous" (1995). Faculty Scholarship. 822. https://researchrepository.wvu.edu/faculty_publications/822 This Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarship by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Cardinal invariants concerning functions whose sum is almost continuous. Krzysztof Ciesielski1, Department of Mathematics, West Virginia University, Mor- gantown, WV 26506-6310 ([email protected]) Arnold W. Miller1, York University, Department of Mathematics, North York, Ontario M3J 1P3, Canada, Permanent address: University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wis- consin 53706-1388, USA ([email protected]) Abstract Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ RR for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F . We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.