DOCSLIB.ORG

Introduction to Abstract Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Abstract Algebra INTRODUCTION TO ABSTRACT ALGEBRA G. JANELIDZE Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch 7701, Cape Town, South Africa Last updated on 20 September 2013 This is an elementary course aiming to introduce very first concepts of abstract algebra for third year students interested in pure mathematics. It is divided into the following (two chapters and) sections: I. First algebraic and related structures 1. Algebraic operations..........................................2 2. Magmas and unitary magmas.....................................3 3. Semigroups....................................................4 4. Monoids.......................................................5 5. Closure operators.............................................5 6. Equivalence relations.........................................7 7. Order relations...............................................8 8. Categories....................................................9 9. Isomorphism..................................................11 10. Initial and terminal objects.................................12 11. Algebras, homomorphisms, isomorphisms........................13 12. Subalgebras..................................................15 13. Products.....................................................16 14. Quotient algebras............................................16 15. Canonical factorization of homomorphisms.....................17 16. Classical algebraic structures...............................18 17. Quotient groups, rings, and modules..........................20 II. Lattices, semirings, number systems, and foundation of linear algebra 1. Commutativity................................................25 2. Semilattices.................................................26 3. Lattices.....................................................29 4. Distributivity and complements...............................31 5. Complete lattices............................................33 6. Boolean algebras.............................................34 7. Semirings and semimodules....................................37 8. The semiring ℕ of natural numbers............................38 9. Number systems, ℤ, ℚ, ℝ, and ℂ as rings, and their modules...40 10. Pointed categories...........................................46 11. Products and coproducts......................................47 12. Direct sums..................................................51 13. Free algebras and free semimodules...........................52 14. Vector spaces................................................55 We shall refer to these sections as follows: say, “Section 10” means “Section 10 in this chapter”, while “Section I.10” means “Section 10 in Chapter I”, when we refer to it in Chapter II. I am grateful to Dr. Amartya Goswami for a number of misprint corrections. 1 I. FIRST ALGEBRAIC AND RELATED STRUCTURES 1. Algebraic operations For a natural number n and a set A, the set of all maps from {1,…,n} to A will be denoted by An; this includes the case n = 0, where {1,…,n} becomes the empty set and therefore An becomes a one-element set. For n = 1, the set An will be identified with A, and for n = 2, 3,…, the elements of An will be written as n-member sequences (a1,…,an) of elements in A. Definition 1.1. For n = 0, 1, 2,…, an n-ary operation on a set A is a map from An to A. We will also use special terms for small n’s: 0-ary nullary 1-ary unary 2-ary binary 3-ary ternary Remarks and Conventions 1.2. (a) Since to give a nullary operation on A is the same as to pick up an element in A, we will simply identify them; accordingly the nullary operations are sometimes called constants. (b) If is an n-ary operation with n = 2, 3,…, we will write (a1,…,an) instead of ((a1,…,an)). Furthermore, for n = 2, instead of (a1,a2) we usually write a1a2, or simply a1a2 – especially when we think of as a kind of multiplication. Binary operations play an especially important role. When the ground set A has a small number of elements, it is convenient to define binary operations on it with tables, such as: a b a a a b a b where A = {a,b} and aa = ab = ba = a, bb = b, or a b a b b b b b where A = {a,b} again, but now aa = ab = ba = bb = b. That is, when A has n elements, the table contains n 1 rows and n 1 columns with the following entries in their cells: the first cell is blank, and the rest of the first row lists the elements of A in any order; the rest of the first column lists the elements of A (preferably) in the same order; 2 for i and j both greater than 1, the cell on the intersection of i-row and j-column has the element ab in it – where a stays in the i-th cell of the first column and b stays in the j-th cell of the first row. Definition 1.3. Let be a binary operation on a set A, and let us write (a,b) = ab, as above. The operation is said to be (a) associative, if a(bc) = (ab)c for all a, b, c in A; (b) commutative, if ab = ba for all a, b in A; (c) idempotent, if aa = a for all a in A. 2. Magmas and unitary magmas Definition 2.1. A magma1 is a pair (M,m), where M is a set and m a binary operation on M. Convention 2.2. Whenever only one magma (M,m) is considered, we will always write m(a,b) = ab and often write just M instead of (M,m). Definition 2.3. An element e in a magma M is said to be: (a) an idempotent, if ee = e; (b) a left identity (or a left unit), if ea = a for every a in M; (c) a right identity (or a right unit), if ae = a for every a in M; (d) an identity2 (or a unit), if it is a left and a right identity at the same time, i.e. if ea = a = ae for every a in M. Theorem 2.3. In an arbitrary magma: (a) every left identity and every right identity is an idempotent; (b) if e is a left identity and e' is a right identity, then e = e'. Proof. (a) is trivial. (b): e = ee' = e'. Remark 2.4. For an arbitrary set M one can define a binary operation on it by ab = b for all a and b in M. In the resulting magma, every element is a left identity. And of course one can do the same with the right identities by putting put ab = a. On the other hand Theorem 2.3(b) tells us that the existence of at least one left identity and at least one right identity in the same magma immediately implies that all of them are equal, yielding a unique identity. Definition 2.5. A unitary magma is a triple (M,e,m), where (M,m) be a magma and e its identity. 1 In old literature magmas were sometimes called groupoids; this has been changed in order to avoid confusions with groupoids in category theory. 2 As follows from 2.3(b) below (see also Remark 2.4), we could also say “the identity”. 3 Convention 2.6. Whenever only one unitary magma (M,e,m) is considered, we will (still use Convention 2.2 and) write 1 instead of e. Remark 2.7. As we see from Remark 2.4, it is a triviality that every unitary magma has exactly one left identity, exactly one right identity, and exactly one identity (all the same). 3. Semigroups Definition 3.1. A magma (M,m) is called a semigroup if m is associative. Theorem 3.2. If (M,m) is a semigroup, then there exists a unique sequence of operations m1, m2, … on M such that: (a) mn is an n-ary operation (n = 1, 2,…); (b) m1(a) = a for every a in M; (c) mp(a1,…,ap)mq(b1,…,bq) = mp+q(a1,…,ap,b1,…,bq) for all p = 1, 2,…; q = 1, 2,…; and a1,…, ap, b1,…, bq in M. Proof. Existence. Let us define m1, m2, … inductively by (b) for m1 and by mn+1(a1,…,an1) = mn(a1,…,an)an1 for m2, m3, …, i.e. for n = 1, 2,…. After this we will prove (c) by induction in q as follows: for q = 1 we have mp(a1,…,ap)mq(b1) = mp(a1,…,ap)b1 = mp1(a1,…,ap,b) for each p; for q > 1, using the inductive assumption and associativity, we then obtain mp(a1,…,ap)mq(b1,…,bq) = mp(a1,…,ap)(mq1(b1,…,bq1)bq) = (mp(a1,…,ap)mq1(b1,…,bq1))bq = mpq1(a1,…,ap,b1,…,bq1)bq = mp+q(a1,…,ap,b1,…,bq), as desired. Uniqueness. From our assumptions on the sequence m1, m2, … we obtain mp+1(a1,…,ap1) = mp(a1,…,ap)m1(ap1) = mp(a1,…,ap)ap1 and so our inductive definition was a consequence of those assumptions. According to Convention 2.2, it is also convenient to avoid writing the letter m for the operations introduced in Theorem 3.2. That is, one writes n n mn(a1,…,an) = a1…an = ai = i=1ai (3.1) i=1 and, in this notation, the formula given in 3.2(c) becomes 4 (a1…ap)(ap1…apq) = a1…apq, (3.2) or, equivalently p pq pq (i=1ai)(i =p1ai) = i =1ai, (3.3) n Furthermore, for a1 = … = an = a, one writes a instead of a1…an, and (3.2) becomes apaq = apq. (3.4) 4. Monoids Definition 4.1. A monoid is a unitary magma (M,e,m), in which m is associative. In other words, a monoid is a unitary magma (M,e,m), in which (M,m), is a semigroup. Theorem 3.2 reformulates for monoids as follows: Theorem 4.2. If (M,e,m) is a monoid, then there exists a unique sequence of operations m0, m1, … on M such that: (a) mn is an n-ary operation (n = 0, 1,…); (b) m1(a) = a for every a in M; (c) mp(a1,…,ap)mq(b1,…,bq) = mp+q(a1,…,ap,b1,…,bq) for all p = 0, 1,…; q = 0, 1,…; and a1,…, ap, b1,…, bq in M. Proof. As follows from Theorem 3.2, all we need is to take care of m0. That is, given a sequence m1, m2, … as in Theorem 3.2, we need to prove that there exists a unique element m0 in M satisfying m0m0 = m0, mp(a1,…,ap)m0 = mp(a1,…,ap), and m0mq(b1,…,bq) = mq(b1,…,bq) for all p = 1, 2,…; q = 1, 2,…; and a1,…, ap, b1,…, bq in M.
Load more

Recommended publications
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Algebras of Boolean Inverse Monoids – Traces and Invariant Means
    C*-algebras of Boolean inverse monoids { traces and invariant means Charles Starling∗ Abstract ∗ To a Boolean inverse monoid S we associate a universal C*-algebra CB(S) and show that it is equal to Exel's tight C*-algebra of S. We then show that any invariant mean on S (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a ∗ trace on CB(S), and vice-versa, under a condition on S equivalent to the underlying ∗ groupoid being Hausdorff. Under certain mild conditions, the space of traces of CB(S) is shown to be isomorphic to the space of invariant means of S. We then use many known results about traces of C*-algebras to draw conclusions about invariant means on Boolean inverse monoids; in particular we quote a result of Blackadar to show that any metrizable Choquet simplex arises as the space of invariant means for some AF inverse monoid S. 1 Introduction This article is the continuation of our study of the relationship between inverse semigroups and C*-algebras. An inverse semigroup is a semigroup S for which every element s 2 S has a unique \inverse" s∗ in the sense that ss∗s = s and s∗ss∗ = s∗: An important subsemigroup of any inverse semigroup is its set of idempotents E(S) = fe 2 S j e2 = eg = fs∗s j s 2 Sg. Any set of partial isometries closed under product and arXiv:1605.05189v3 [math.OA] 19 Jul 2016 involution inside a C*-algebra is an inverse semigroup, and its set of idempotents forms a commuting set of projections.
    [Show full text]
  • On the Representation of Boolean Magmas and Boolean Semilattices
    Chapter 1 On the representation of Boolean magmas and Boolean semilattices P. Jipsen, M. E. Kurd-Misto and J. Wimberley Abstract A magma is an algebra with a binary operation ·, and a Boolean magma is a Boolean algebra with an additional binary operation · that distributes over all finite Boolean joins. We prove that all square-increasing (x ≤ x2) Boolean magmas are embedded in complex algebras of idempotent (x = x2) magmas. This solves a problem in a recent paper [3] by C. Bergman. Similar results are shown to hold for commutative Boolean magmas with an identity element and a unary inverse operation, or with any combination of these properties. A Boolean semilattice is a Boolean magma where · is associative, com- mutative, and square-increasing. Let SL be the class of semilattices and let S(SL+) be all subalgebras of complex algebras of semilattices. All members of S(SL+) are Boolean semilattices and we investigate the question of which Boolean semilattices are representable, i.e., members of S(SL+). There are 79 eight-element integral Boolean semilattices that satisfy a list of currently known axioms of S(SL+). We show that 72 of them are indeed members of S(SL+), leaving the remaining 7 as open problems. 1.1 Introduction The study of complex algebras of relational structures is a central part of algebraic logic and has a long history. In the classical setting, complex al- gebras connect Kripke semantics for polymodal logics to Boolean algebras with normal operators. Several varieties of Boolean algebras with operators are generated by the complex algebras of a standard class of relational struc- tures.
    [Show full text]
  • Binary Opera- Tions, Magmas, Monoids, Groups, Rings, fields and Their Homomorphisms
    1. Introduction In this chapter, I introduce some of the fundamental objects of algbera: binary opera- tions, magmas, monoids, groups, rings, fields and their homomorphisms. 2. Binary Operations Definition 2.1. Let M be a set. A binary operation on M is a function · : M × M ! M often written (x; y) 7! x · y. A pair (M; ·) consisting of a set M and a binary operation · on M is called a magma. Example 2.2. Let M = Z and let + : Z × Z ! Z be the function (x; y) 7! x + y. Then, + is a binary operation and, consequently, (Z; +) is a magma. Example 2.3. Let n be an integer and set Z≥n := fx 2 Z j x ≥ ng. Now suppose n ≥ 0. Then, for x; y 2 Z≥n, x + y 2 Z≥n. Consequently, Z≥n with the operation (x; y) 7! x + y is a magma. In particular, Z+ is a magma under addition. Example 2.4. Let S = f0; 1g. There are 16 = 42 possible binary operations m : S ×S ! S . Therefore, there are 16 possible magmas of the form (S; m). Example 2.5. Let n be a non-negative integer and let · : Z≥n × Z≥n ! Z≥n be the operation (x; y) 7! xy. Then Z≥n is a magma. Similarly, the pair (Z; ·) is a magma (where · : Z×Z ! Z is given by (x; y) 7! xy). Example 2.6. Let M2(R) denote the set of 2 × 2 matrices with real entries. If ! ! a a b b A = 11 12 , and B = 11 12 a21 a22 b21 b22 are two matrices, define ! a b + a b a b + a b A ◦ B = 11 11 12 21 11 12 12 22 : a21b11 + a22b21 a21b12 + a22b22 Then (M2(R); ◦) is a magma.
    [Show full text]
  • Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
    Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Loop Near-Rings and Unique Decompositions of H-Spaces
    Algebraic & Geometric Topology 16 (2016) 3563–3580 msp Loop near-rings and unique decompositions of H-spaces DAMIR FRANETICˇ PETAR PAVEŠIC´ For every H-space X , the set of homotopy classes ŒX; X possesses a natural al- gebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull–Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper, we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull–Schmidt theory for infinite products. Finally, we show that indecomposable finite p–local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray. 55P45; 16Y30 Introduction In this paper, we discuss relations between unique decomposition theorems in alge- bra and homotopy theory. Unique decomposition theorems usually state that sum or product decompositions (depending on the category) whose factors are strongly indecomposable are essentially unique. The standard algebraic example is the Krull– Schmidt–Remak–Azumaya theorem. In its modern form, the theorem states that any decomposition of an R–module into a direct sum of indecomposable modules is unique, provided that the endomorphism rings of the summands are local rings; see Facchini [8, Theorem 2.12].
    [Show full text]
  • Monoid Modules and Structured Document Algebra (Extendend Abstract)
    Monoid Modules and Structured Document Algebra (Extendend Abstract) Andreas Zelend Institut fur¨ Informatik, Universitat¨ Augsburg, Germany [email protected] 1 Introduction Feature Oriented Software Development (e.g. [3]) has been established in computer science as a general programming paradigm that provides formalisms, meth- ods, languages, and tools for building maintainable, customisable, and exten- sible software product lines (SPLs) [8]. An SPL is a collection of programs that share a common part, e.g., functionality or code fragments. To encode an SPL, one can use variation points (VPs) in the source code. A VP is a location in a pro- gram whose content, called a fragment, can vary among different members of the SPL. In [2] a Structured Document Algebra (SDA) is used to algebraically de- scribe modules that include VPs and their composition. In [4] we showed that we can reason about SDA in a more general way using a so called relational pre- domain monoid module (RMM). In this paper we present the following extensions and results: an investigation of the structure of transformations, e.g., a condi- tion when transformations commute, insights into the pre-order of modules, and new properties of predomain monoid modules. 2 Structured Document Algebra VPs and Fragments. Let V denote a set of VPs at which fragments may be in- serted and F(V) be the set of fragments which may, among other things, contain VPs from V. Elements of F(V) are denoted by f1, f2,... There are two special elements, a default fragment 0 and an error . An error signals an attempt to assign two or more non-default fragments to the same VP within one module.
    [Show full text]
  • Arxiv:Math/0310146V1 [Math.AT] 10 Oct 2003 Usinis: Question Fr Most Aut the the of Algebra”
    MORITA THEORY IN ABELIAN, DERIVED AND STABLE MODEL CATEGORIES STEFAN SCHWEDE These notes are based on lectures given at the Workshop on Structured ring spectra and their applications. This workshop took place January 21-25, 2002, at the University of Glasgow and was organized by Andy Baker and Birgit Richter. Contents 1. Introduction 1 2. Morita theory in abelian categories 2 3. Morita theory in derived categories 6 3.1. The derived category 6 3.2. Derived equivalences after Rickard and Keller 14 3.3. Examples 19 4. Morita theory in stable model categories 21 4.1. Stable model categories 22 4.2. Symmetric ring and module spectra 25 4.3. Characterizing module categories over ring spectra 32 4.4. Morita context for ring spectra 35 4.5. Examples 38 References 42 1. Introduction The paper [Mo58] by Kiiti Morita seems to be the first systematic study of equivalences between module categories. Morita treats both contravariant equivalences (which he calls arXiv:math/0310146v1 [math.AT] 10 Oct 2003 dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’). The term ‘Morita theory’ is now used for results concerning equivalences of various kinds of module categories. The authors of the obituary article [AGH] consider Morita’s theorem “probably one of the most frequently used single results in modern algebra”. In this survey article, we focus on the covariant form of Morita theory, so our basic question is: When do two ‘rings’ have ‘equivalent’ module categories ? We discuss this question in different contexts: • (Classical) When are the module categories of two rings equivalent as categories ? Date: February 1, 2008.
    [Show full text]
  • Programming with Algebraic Structures: Design of the Magma Language
    Programming with Algebraic Structures: Design of the Magma Language Wieb Bosma John Cannon Graham Matthews School of Mathematics and Statistics, University of Sydney Sydney, iVSW 2006, Australia Abstract classifying all structures that satisfy some (interesting) set of axioms. Notable successes of this approach have MAGMA is a new software system for computational been the classification of all simple Lie algebras over the algebra, number theory and geometry whose design is field of complex numbers, the Wedderburn classification centred on the concept of algebraic structure (magma). of associative algebras, and the very recent classification The use of algebraic structure as a design paradigm of finite simple groups. Classification problems t ypically provides a natural strong typing mechanism. Further, concern themselves with categories of algebraic struc- structures and their morphisms appear in the language tures, and their solution usually requires detailed anal- as first class objects. Standard mathematical notions ysis of entire structures, rather then just consideration are used for the basic data types. The result is a power- of their elements, that is, second order computation. ful, clean language which deals with objects in a math- Examination of the major computer algebra systems ematically rigorous manner. The conceptual and im- reveals that most of them were designed around the no- plementation ideas behind MAGMA will be examined in tion of jirst order computation. Further, systems such this paper. This conceptual base differs significantly as Macsyma, Reduce, Maple [6] and Mathematical [13] from those underlying other computer algebra systems. assume that all algebraic objects are elements of one of a small number of elementary structures.
    [Show full text]