Unit-IV : Algebraic Structures
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Algebra I (Math 200)
Algebra I (Math 200) UCSC, Fall 2009 Robert Boltje Contents 1 Semigroups and Monoids 1 2 Groups 4 3 Normal Subgroups and Factor Groups 11 4 Normal and Subnormal Series 17 5 Group Actions 22 6 Symmetric and Alternating Groups 29 7 Direct and Semidirect Products 33 8 Free Groups and Presentations 35 9 Rings, Basic Definitions and Properties 40 10 Homomorphisms, Ideals and Factor Rings 45 11 Divisibility in Integral Domains 55 12 Unique Factorization Domains (UFD), Principal Ideal Do- mains (PID) and Euclidean Domains 60 13 Localization 65 14 Polynomial Rings 69 Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) S together with a binary operation ∗ is called a semigroup, if ∗ is as- sociative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp. -
Algebraic Structure of Genetic Inheritance 109
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 2, April 1997, Pages 107{130 S 0273-0979(97)00712-X ALGEBRAIC STRUCTURE OF GENETIC INHERITANCE MARY LYNN REED Abstract. In this paper we will explore the nonassociative algebraic struc- ture that naturally occurs as genetic information gets passed down through the generations. While modern understanding of genetic inheritance initiated with the theories of Charles Darwin, it was the Augustinian monk Gregor Mendel who began to uncover the mathematical nature of the subject. In fact, the symbolism Mendel used to describe his first results (e.g., see his 1866 pa- per Experiments in Plant-Hybridization [30]) is quite algebraically suggestive. Seventy four years later, I.M.H. Etherington introduced the formal language of abstract algebra to the study of genetics in his series of seminal papers [9], [10], [11]. In this paper we will discuss the concepts of genetics that suggest the underlying algebraic structure of inheritance, and we will give a brief overview of the algebras which arise in genetics and some of their basic properties and relationships. With the popularity of biologically motivated mathematics continuing to rise, we offer this survey article as another example of the breadth of mathematics that has biological significance. The most com- prehensive reference for the mathematical research done in this area (through 1980) is W¨orz-Busekros [36]. 1. Genetic motivation Before we discuss the mathematics of genetics, we need to acquaint ourselves with the necessary language from biology. A vague, but nevertheless informative, definition of a gene is simply a unit of hereditary information. -
Boolean and Abstract Algebra Winter 2019
Queen's University School of Computing CISC 203: Discrete Mathematics for Computing II Lecture 7: Boolean and Abstract Algebra Winter 2019 1 Boolean Algebras Recall from your study of set theory in CISC 102 that a set is a collection of items that are related in some way by a common property or rule. There are a number of operations that can be applied to sets, like [, \, and C. Combining these operations in a certain way allows us to develop a number of identities or laws relating to sets, and this is known as the algebra of sets. In a classical logic course, the first thing you typically learn about is propositional calculus, which is the branch of logic that studies propositions and connectives between propositions. For instance, \all men are mortal" and \Socrates is a man" are propositions, and using propositional calculus, we may conclude that \Socrates is mortal". In a sense, propositional calculus is very closely related to set theory, in that propo- sitional calculus is the study of the set of propositions together with connective operations on propositions. Moreover, we can use combinations of connective operations to develop the laws of propositional calculus as well as a collection of rules of inference, which gives us even more power to manipulate propositions. Before we continue, it is worth noting that the operations mentioned previously|and indeed, most of the operations we have been using throughout these notes|have a special name. Operations like [ and \ apply to pairs of sets in the same way that + and × apply to pairs of numbers. -
MGF 1106 Learning Objectives
Learning Objectives L01 Set Concepts and Subsets (2.1, 2.2) 1. Indicate sets by the description method, roster method, and by using set builder notation. 2. Determine if a set is well defined. 3. Determine if a set is finite or infinite. 4. Determine if sets are equal, equivalent, or neither. 5. Find the cardinal number of a set. 6. Determine if a set is equal to the empty set. 7. Use the proper notation for the empty set. 8. Give an example of a universal set and list all of its subsets and all of its proper subsets. 9. Properly use notation for element, subset, and proper subset. 10. Determine if a number represents a cardinal number or an ordinal number. 11. Be able to determine the number of subsets and proper subsets that can be formed from a universal set without listing them. L02 Venn Diagrams and Set Operations (2.3, 2.4) 1. Determine if sets are disjoint. 2. Find the complement of a set 3. Find the intersection of two sets. 4. Find the union of two sets. 5. Find the difference of two sets. 6. Apply several set operations involved in a statement. 7. Determine sets from a Venn diagram. 8. Use the formula that yields the cardinal number of a union. 9. Construct a Venn diagram given two sets. 10. Construct a Venn diagram given three sets. L03 Equality of Sets; Applications of Sets (2.4, 2.5) 1. Determine if set statements are equal by using Venn diagrams or DeMorgan's laws. -
On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES . -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
Instructional Guide for Basic Mathematics 1, Grades 10 To
REPORT RESUMES ED 016623 SE 003 950 INSTRUCTIONAL GUIDE FOR BASIC MATHEMATICS 1,GRADES 10 TO 12. BY- RICHMOND, RUTH KUSSMANN LOS ANGELES CITY SCHOOLS, CALIF. REPORT NUMBER X58 PUB DATE 66 EDRS PRICE MF -$0.25 HC-61.44 34P. DESCRIPTORS- *CURRICULUM DEVELOPMENT,*MATHEMATICS, *SECONDARY SCHOOL MATHEMATICS, *TEACHING GUIDES,ARITHMETIC, COURSE CONTENT, GRADE 10, GRADE 11, GRADE 12,GEOMETRY, LOW ABILITY STUDENTS, SLOW LEARNERS, STUDENTCHARACTERISTICS, LOS ANGELES, CALIFORNIA, THIS INSTRUCTIONAL GUIDE FORMATHEMATICS 1 OUTLINES CONTENT AND PROVIDES TEACHINGSUGGESTIONS FOR A FOUNDATION COURSE FOR THE SLOW LEARNER IN THESENIOR HIGH SCHOOL. CONSIDERATION HAS BEEN GIVEN IN THEPREPARATION OF THIS DOCUMENT TO THE STUDENT'S INTERESTLEVELS AND HIS ABILITY TO LEARN. THE GUIDE'S PURPOSE IS TOENABLE THE STUDENTS TO UNDERSTAND AND APPLY THE FUNDAMENTALMATHEMATICAL ALGORITHMS AND TO ACHIEVE SUCCESS ANDENJOYMENT IN WORKING WITH MATHEMATICS. THE CONTENT OF EACH UNITINCLUDES (1) DEVELOPMENT OF THE UNIT, (2)SUGGESTED TEACHING PROCEDURES, AND (3) STUDENT EVALUATION.THE MAJOR PORTION OF THE MATERIAL IS DEVOTED TO THE FUNDAMENTALOPERATIONS WITH WHOLE NUMBERS. IDENTIFYING AND CLASSIFYING ELEMENTARYGEOMETRIC FIGURES ARE ALSO INCLUDED. (RP) WELFARE U.S. DEPARTMENT OFHEALTH, EDUCATION & OFFICE OF EDUCATION RECEIVED FROM THE THIS DOCUMENT HAS BEENREPRODUCED EXACTLY AS POINTS OF VIEW OROPINIONS PERSON OR ORGANIZATIONORIGINATING IT. OF EDUCATION STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE POSITION OR POLICY. INSTRUCTIONAL GUIDE BASICMATHEMATICS I GRADES 10 to 12 LOS ANGELES CITY SCHOOLS Division of Instructional Services Curriculum Branch Publication No. X-58 1966 FOREWORD This Instructional Guide for Basic Mathematics 1 outlines content and provides teaching suggestions for a foundation course for the slow learner in the senior high school. -
An Elementary Approach to Boolean Algebra
Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties
A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields). -
Algebraic Structures Lecture 18 Thursday, April 4, 2019 1 Type
Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages Algebraic structures Lecture 18 Thursday, April 4, 2019 In abstract algebra, algebraic structures are defined by a set of elements and operations on those ele- ments that satisfy certain laws. Some of these algebraic structures have interesting and useful computa- tional interpretations. In this lecture we will consider several algebraic structures (monoids, functors, and monads), and consider the computational patterns that these algebraic structures capture. We will look at Haskell, a functional programming language named after Haskell Curry, which provides support for defin- ing and using such algebraic structures. Indeed, monads are central to practical programming in Haskell. First, however, we consider type constructors, and see two new type constructors. 1 Type constructors A type constructor allows us to create new types from existing types. We have already seen several different type constructors, including product types, sum types, reference types, and parametric types. The product type constructor × takes existing types τ1 and τ2 and constructs the product type τ1 × τ2 from them. Similarly, the sum type constructor + takes existing types τ1 and τ2 and constructs the product type τ1 + τ2 from them. We will briefly introduce list types and option types as more examples of type constructors. 1.1 Lists A list type τ list is the type of lists with elements of type τ. We write [] for the empty list, and v1 :: v2 for the list that contains value v1 as the first element, and v2 is the rest of the list. We also provide a way to check whether a list is empty (isempty? e) and to get the head and the tail of a list (head e and tail e).