DOCSLIB.ORG

Everything You Always Wanted to Know About Rings in GAP∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Everything You Always Wanted to Know About Rings in GAP∗ Everything you always wanted to know about rings in GAP∗. (∗ but were afraid to ask) Diplomarbeit zur Erlangung des akademischen Grades eines Diplom-Ingenieurs in der Studienrichtung Technische Mathematik, Stzw. Informations- und Datenverarbeitung Angefertigt am Institut f¨urMathematik der Technisch- Naturwissenschaftlichen Fakult¨atder Johannes Kepler Universit¨atLinz Eingereicht bei Univ.-Prof. Dr. G¨unter Pilz J¨urgenEcker October 7, 1999 Preface We dance around in a ring and suppose, But the secret sits in the middle and knows. Robert Frost (1875{1963) There exist thousands of books on ring-theory each containing hundreds of the- orems, most of them for special classes of rings. Nonetheless given a ring certain properties can not be checked by hand in a reasonable time (unless by a genius). This makes computer programs desireable that do the dirty work. We shall dance around a little and gether the most common results in ring theory. Chapter 1 starts with a glance at universal algebra. In chapter 2 we gether results about rings. Chapters 3 and 4 are about analyzing the structure of a ring and end with the famous density theorem due to Jacobson. Chapters 5 and 6 shall make computation easier in many cases. In chapter 7 we give an overview about results for polynomial rings featuring Buchberger's Algorithm for Groebner Bases. We omit proofs if they are long and not of particular importance for the algorithms. However, the main part is a library of GAP-functions presented in chapter 8. This is a users manual for all the functions for rings so far implemented in GAP. Note, that this is not a collection of super-fast algorithms for very special rings, but of default functions which will work for any ring using as little knowledge about the ring as possible. For those interested in curious programming styles we have added the source code of all new functions in chapter 10. 1 Contents 1 Universal Algebra 8 1.1 Algebras . 8 1.2 Moore{Systems . 10 1.3 More about Congruences and Subdirect Products . 11 1.4 Chain Conditions . 12 2 Rings 13 2.1 Rings and Subrings . 13 2.2 Ideals and Factor Rings . 15 2.3 Modules . 18 3 Radical-Theory 22 3.1 Jacobson Radical . 22 3.2 Nil Radicals . 23 2 CONTENTS 3 4 Subdirect Decomposition 24 4.1 Subdirect Decomposition of Rings . 24 4.2 Consequences of Birkhoff's theorem . 26 4.2.1 Primitive Rings . 26 4.2.2 Semi-simplicity . 28 4.2.3 Density Theorems . 29 5 Chain Conditions 31 5.1 DCC . 31 5.2 ACC . 34 6 Some results for improved algorithms 35 6.1 Ideal Generators . 35 6.2 Factor2ring{Avoidance . 37 6.3 Divide et Impera . 38 7 Polynomial rings 41 7.1 Orderings on R[X].......................... 42 7.2 Reduction . 44 7.3 Reduction Relations . 45 7.4 Reduction in K[X].......................... 46 7.5 Buchberger's Theorem . 47 7.6 Improvements on Buchberger's Algorithm . 49 7.7 Applications . 51 CONTENTS 4 8 GAP{Manual: Rings 53 8.1 Preliminaries . 53 8.1.1 Data Structures . 54 8.1.2 Operations . 54 8.1.3 Algebraic Domains . 55 8.1.4 Elements of Special Classes of Fields . 55 8.2 Defining a Ring . 55 8.2.1 Ring Records . 56 8.2.2 Ring . 58 8.2.3 PolynomialRing . 58 8.2.4 AsRing . 58 8.2.5 AsGroupElement . 59 8.2.6 AsRingElement . 60 8.2.7 DirectProduct . 60 8.3 Subrings and Ideals . 60 8.3.1 Subring . 60 8.3.2 IsSubring . 61 8.3.3 Ideal . 61 8.3.4 IsIdeal . 62 8.3.5 LeftIdeal . 62 8.3.6 IsLeftIdeal . 62 8.3.7 RightIdeal . 63 8.3.8 IsRightIdeal . 63 CONTENTS 5 8.3.9 Operations for (left-, right-) ideals . 63 8.3.10 Rad . 63 8.3.11 IsModularLeftIdeal . 64 8.3.12 IsMaximalIdeal . 64 8.3.13 IsMaximalLeftIdeal . 64 8.3.14 IsMaximalRightIdeal . 64 8.3.15 IsPrimeIdeal . 65 8.3.16 QuickIdealElements . 65 8.3.17 QuickLeftIdealElements . 65 8.3.18 QuickRightIdealElements . 65 8.3.19 Subrings . 66 8.3.20 Ideals . 66 8.3.21 LeftIdeals . 66 8.3.22 RightIdeals . 67 8.3.23 IsSimple . 67 8.4 Factor Rings . 67 8.4.1 FactorRing . 67 8.4.2 IsFactorRing . 68 8.4.3 FactorRingElement . 68 8.5 Decomposition of Rings . 69 8.5.1 SubdirectDecomposition . 69 8.5.2 PartialDecomposition . 71 8.6 Radical Theory . 71 CONTENTS 6 8.6.1 IsNilpotentElement . 72 8.6.2 Nilpotents . 72 8.6.3 IsNilIdeal . 72 8.6.4 Nilradical . 73 8.6.5 JacobsonRadical . 73 8.6.6 IsJacobsonRadicalRing . 73 8.6.7 IsSemisimple . 73 8.7 Polynomials . 74 8.7.1 Variables . 74 8.7.2 Lpp . 74 8.7.3 ReducegModf . 75 8.7.4 NormalFormOfgModF . 75 8.7.5 GroebnerBasis . 76 8.8 Miscellaneous . 76 8.8.1 RingOperationTables . 77 8.8.2 Call . 77 8.9 Deja vu . 78 8.9.1 in . 78 8.9.2 Elements . 78 8.9.3 Others . 78 CONTENTS 7 A Algebraic details 81 A.1 The Abstract Definition of a Ring . 81 A.2 Ordering . 82 A.3 Zorn's Lemma . 82 A.4 Group . 83 B Source Code 84 B.1 The Source File ringplus ...................... 84 B.2 The Source File dispatch.l ..................... 85 B.3 The Source File factor.l ...................... 98 B.4 The Source File subdec.l . 101 B.5 The Source File radical.l . 103 B.6 The Source File misc.l . 122 B.7 The Source File ideals.l . 128 Chapter 1 Universal Algebra Nothing will come of nothing. William Shakespeare (1564{1616) Algebraic structures as semigroups, groups, rings, nearrings, etc. have nice prop- erties in common. Therefore it seems useful to define certain things generally and use the results in each of the various structures. 1.1 Algebras Definition 1 An algebra is a tuple A (A; 1;:::; m;?1;:::;?n; Ω), where A is a set, 1;:::; m are binary operations ◦ ◦ ◦ ◦ on A and ?1;:::;?n are external operations on A with operator domain Ω. τ := (m; n; Ω) is called the type of . If n = 0 we write (m) instead of (m; 0; ).A subset of the class of all algebrasA of the same type is called a class of algebras.; Definition 2 Let λ : λ Λ be a family of algebras of the same type fA 2 g (n; m; Ω) and the same class, where the binary (external) operations of λ are λ λ λ λ A 1 ;:::; n (?1 ;:::;?m). On the cartesian product of the sets Ai we can de- fine◦ an◦ algebra of the same class, defining as binary (external) operations the "componentwise operations": λ (aλ)λ Λ i (bλ)λ Λ := (aλ i bλ)λ Λ 2 2 2 ◦ ◦λ !?j (aλ)λ Λ := (!?j aλ)λ Λ 2 2 for all i 1; : : : ; n , j 1; : : : ; m . 2 f g 2 f g This algebra is called the (direct) product of the algebras Aλ, denoted λ Λ λ. Q 2 A 8 CHAPTER 1. UNIVERSAL ALGEBRA 9 Definition 3 Let , the algebra above, be of class . Then B A together with all operationsA of forms an algebra of theC same class if⊆B is closed under all operations ofA . is called a B-subalgebra of . We denote this or without an indexA B if it's clear fromC the context. A B ≤C A Definition 4 The set (: : : ; aλ;:::): only finitely many aλ = 0 induces a sub- f 6 g algebra of λ Λ λ, called the direct sum of the λ. We write λ Λ λ. Q 2 A A L 2 A Definition 5 Let S be a set, a binary operation on S, ? an external operation on S with operator domain Ω◦and a relation in S. is compatible with iff a; b; a;~ ~b.
Load more

Recommended publications
  • 2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem
    2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem by Ian Payne A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Pure Mathematics Waterloo, Ontario, Canada, 2017 c Ian Payne 2017 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner George McNulty Professor University of South Carolina Supervisor Ross Willard Professor University of Waterloo Internal Member Stanley Burris Distinguished Professor Emeritus Adjunct Professor University of Waterloo Internal Member Rahim Moosa Professor University of Waterloo Internal-external Member Chris Godsil Professor University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Semilattices are algebras known to have an important connection to partially ordered sets. In particular, if a partially ordered set (A; ≤) has greatest lower bounds, a semilattice (A; ^) can be associated to the order where a ^ b is the greatest lower bound of a and b. In this thesis, we study a class of algebras known as 2-semilattices, which is a generalization of the class of semilattices. Similar to the correspondence between partial orders and semilattices, there is a correspondence between certain digraphs and 2-semilattices. That is, to every 2-semilattice, there is an associated digraph which holds information about the 2-semilattice.
    [Show full text]
  • Locally Compact Vector Spaces and Algebras Over Discrete Fieldso
    LOCALLY COMPACT VECTOR SPACES AND ALGEBRAS OVER DISCRETE FIELDSO BY SETH WARNER The structure of locally compact vector spaces over complete division rings topologized by a proper absolute value (or, more generally, over complete division rings of type V) is summarized in the now classical theorem that such spaces are necessarily finite-dimensional and are topologically isomorphic to the cartesian product of a finite number of copies of the scalar division ring [9, Theorems 5 and 7], [6, pp. 27-31]. Here we shall continue a study of locally compact vector spaces and algebras over infinite discrete fields that was initiated in [16]. Since any topological vector space over a topological division ring K remains a topological vector space if K is retopologized with the discrete topology, some restriction needs to be imposed in our investigation, and the most natural restriction is straightness : A topological vector space F over a topological division ring K is straight if for every nonzero ae E,fa: X -^ Xa is a homeomorphism from K onto the one-dimensional subspace generated by a. Thus if K is discrete, a straight A^-vector space is one all of whose one-dimensional subspaces are discrete. Any Hausdorff vector space over a division ring topologized by a proper absolute value is straight [6, Proposition 2, p. 25]. In §1 we shall prove that if F is an indiscrete straight locally compact vector space over an infinite discrete division ring K, then K is an absolutely algebraic field of prime characteristic and [F : K] > X0. In §2 we shall see that finite- dimensional subspaces of straight locally compact vector spaces need not be discrete.
    [Show full text]
  • GENERATING SUBDIRECT PRODUCTS 1. Introduction A
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository GENERATING SUBDIRECT PRODUCTS PETER MAYR AND NIK RUSKUCˇ Abstract. We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence per- mutable varieties, which generalise previously known results for groups, and which apply to modules, rings, K-algebras and loops. For instance, if C is a fiber product of A and B over a common quotient D, and if A, B and D are finitely presented, then C is finitely generated. For subdirect products of more than two factors we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and K-algebras. In particular, let C be a subdirect product of K- algebras A1;:::;An for a Noetherian ring K such that the projection of C onto any Ai × Aj has finite co-rank in Ai × Aj. Then C is finitely generated (resp. finitely presented) if and only if all Ai are finitely generated (resp. finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties. 1. Introduction A subdirect product of algebraic structures A and B is a subalgebra C of the direct product A × B such that the projections of A × B onto A and B both remain surjective when restricted to C.
    [Show full text]
  • Subgroups of Direct Products of Limit Groups
    ANNALS OF MATHEMATICS Subgroups of direct products of limit groups By Martin R. Bridson, James Howie, Charles F. Miller III, and Hamish Short SECOND SERIES, VOL. 170, NO. 3 November, 2009 anmaah Annals of Mathematics, 170 (2009), 1447–1467 Subgroups of direct products of limit groups By MARTIN R. BRIDSON, JAMES HOWIE, CHARLES F. MILLER III, and HAMISH SHORT Dedicated to the memory of John Stallings Abstract If 1; : : : ; n are limit groups and S 1 n is of type FPn.Q/ then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela. 1. Introduction The systematic study of the higher finiteness properties of groups was initiated forty years ago by Wall [28] and Serre [25]. In 1963, Stallings [27] constructed the first example of a finitely presented group with infinite-dimensional H3. Q/; I his example was a subgroup of a direct product of three free groups. This was the first indication of the great diversity to be found amongst the finitely presented subgroups of direct products of free groups, a theme developed in [5]. In contrast, Baumslag and Roseblade [3] proved that in a direct product of two free groups the only finitely presented subgroups are the obvious ones: such a subgroup is either free or has a subgroup of finite index that is a direct product of free groups. In [10] the present authors explained this contrast by proving that the exotic behaviour among the finitely presented subgroups of direct products of free groups is accounted for entirely by the failure of higher homological-finiteness conditions.
    [Show full text]
  • On Subdirect Factors of Projective Modules and Applications to System
    ON SUBDIRECT FACTORS OF A PROJECTIVE MODULE AND APPLICATIONS TO SYSTEM THEORY MOHAMED BARAKAT ABSTRACT. We extend a result of NAPP AVELLI, VAN DER PUT, and ROCHA with a system- theoretic interpretation to the noncommutative case: Let P be a f.g. projective module over a two- sided NOETHERian domain. If P admits a subdirect product structure of the form P =∼ M ×T N over a factor module T of grade at least 2 then the torsion-free factor of M (resp. N) is projective. 1. INTRODUCTION This paper provides two homologically motivated generalizations of a module-theoretic result proved by NAPP AVELLI, VAN DER PUT, and ROCHA. This result was expressed in [NAR10] using the dual system-theoretic language and applied to behavioral control. Their algebraic proof covers at least the polynomial ring R = k[x1,...,xn] and the LAURENT polynomial ring R = ± ± k[x1 ,...,xn ] over a field k. The corresponding module-theoretic statement is the following: Theorem 1.1. Let R be one of the above rings and M = Rq/A a finitely generated torsion- free module. If there exists a submodule B ≤ Rq such that A ∩ B =0 and T := Rq/(A + B) is of codimension at least 2 then M is free. In fact, they prove a more general statement of which the previous is obviously a special case. However, the special statement implies the more general one. Theorem 1.2 ([NAR10, Theorem 18]). Let R be one of the above rings and M = Rq/A a finitely generated module. If there exists a submodule B ≤ Rq such that A ∩ B = 0 and T := Rq/(A + B) is of codimension at least 2 then the torsion-free factor M/ t(M) of M is free.
    [Show full text]
  • Algebraic Structures from Quantum and Fuzzy Logics
    Universit`adegli studi di Cagliari Dipartimento di Pedagogia, Psicologia e Filosofia Dottorato in Storia, Filosofia e Didattica delle Scienze Coordinatore: Prof. Marco Giunti Ciclo XXVIII S.S.D: M-FIL 02, MAT 01 Algebraic structures from quantum and fuzzy logics Candidate: Stefano Bonzio Thesis advisor: Prof. Francesco Paoli Research supervisor: Prof. Antonio Ledda Esame finale anno accademico 2014/2015 Contents Abstract 9 Introduction 11 1 Mathematical Background 13 1.1 Preliminaries . 13 1.2 Lattice theory and universal algebra . 16 1.2.1 Algebras . 16 1.2.2 Fundamentals of lattice theory . 20 1.2.3 Subalgebras, quotients, direct and subdirect products . 22 1.2.4 Class operators, varieties and congruence properties. 27 1.3 Basic of Group theory . 30 1.3.1 Group actions . 33 1.4 Basics of fuzzy and quantum structures . 33 1.4.1 MV-algebras . 34 1.4.2 Quantum structures and Basic algebras . 35 1.5 Residuated Lattices . 37 1.6 Directoids . 40 2 Orthogonal relational systems 43 2.1 Relational systems with involution . 44 2.2 Central elements and decomposition . 53 2.3 Amalgamation property . 59 3 Relational structures and residuation 63 3.1 Residuated relational systems . 63 3.2 Pre-ordered residuated systems . 65 3.3 Residuated directoids . 73 3 4 Quantum structures as near semirings 81 4.1 Near semirings . 82 4.2 Basic algebras as near semirings . 90 4.3 Congruence Properties ofLukasiewicz near semirings . 94 4.4 Orthomodular lattices as near semirings . 96 4.5 Central elements and decomposition . 101 4.6 Appendix on central elements . 110 5 Appendix: extensions of the Rubik's Cube 113 5.1 A group theoretical approach to the Rubik's Cube .
    [Show full text]
  • On Normal Subgroups of Direct Products
    Proceedings of the Edinburgh Mathematical Society (1990) 33, 309-319 © ON NORMAL SUBGROUPS OF DIRECT PRODUCTS by F. E. A. JOHNSON (Received 10th April 1989) We investigate the equivalence classes of normal subdirect products of a product of free groups F., x • • • x Fnit under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when fcS3, and list the finite number of classes when k = 2. 1980 Mathematics subject classification (1985 Revision): 2OEO7; 20E36. 0. Introduction Two subgroups Hx, H2 of a group G are said to be conjugate in the generalised sense when tx(Hl) = H2 for some automorphism a of G. Similarly, we may consider generalised commensurability, by which we mean that cc(H^) n H2 has finite index in both a^H^) and H2. In this paper, we investigate the generalised conjugacy and commensurability classes of normal subgroups in a direct product of free groups; let Fk denote the free group with basis (X,-)^.-^, and for O^d^min {n,m}, let N(n,m,d) be the subgroup of FnxFm (Xh 1) generated by (Xipl) l^i<j^ 1 1 where Xi} is the commutator Xij=XiXJXj~ XJ . Then N(n,m,d) is a finitely generated normal subgroup, with infinite index when d^.\; it is, moreover, a subdirect product; that is, it projects epimorphically onto each factor. We will show that, up to generalised commensurability, these are the only normal subdirect products.
    [Show full text]
  • 1 Motivation 2 Modules and Primitive Rings
    My Favorite Theorem Kiddie; October 23, 2017 Yuval Wigderson Remember! Not all rings are commutative! 1 Motivation Definition. A ring R is called Boolean if all its elements are idempotents: for every a 2 R, a2 = a. Proposition. All Boolean rings are commutative. Proof. Suppose R is Boolean. We can write 1 + 1 = (1 + 1)2 = 12 + 12 + 12 + 12 = 1 + 1 + 1 + 1 which implies that 1 = −1 (i.e. R has characteristic 2). Now, fix a; b 2 R. Then a + b = (a + b)2 = a2 + ab + ba + b2 = a + ab + ba + b Subtracting a + b from both sides gives ab = −ba, and since 1 = −1, we get that ab = ba. Since this holds for all a; b 2 R, we see that R is commutative. For an interesting extension of this theorem, call a ring R troolean if a3 = a for all a 2 R. Then we have the following result: Proposition. All troolean rings are commutative. This can also be proved by similar elementary methods, though it's significantly trickier. However, both of these reults are special cases of the following massive generalization, which is both my favorite theorem and the subject of this talk: Theorem (Jacobson's Commutativity Theorem). Let R be a ring, and suppose that it satisfies Jacobson's condition: for every a 2 R, there exists some integer n(a) ≥ 2 with an(a) = a. Then R is commutative. It's worth pondering this theorem|I find it deeply surprising, since it feels like powers should have very little to do with commutativity.
    [Show full text]
  • The Congruences of a Finite Lattice a Proof-By-Picture Approach
    The Congruences of a Finite Lattice A Proof-by-Picture Approach Second Edition George Gr¨atzer arXiv:2104.06539v1 [math.RA] 13 Apr 2021 Birkh¨auserVerlag, New York @ 2016 Birkh¨auserVerlag To L´aszl´oFuchs, my thesis advisor, my teacher, who taught me to set the bar high; and to my coauthors, who helped me raise the bar. Short Contents Table of Notation xiii Picture Gallery xvii Preface to the Second Edition xix Introduction xxi I A Brief Introduction to Lattices 1 1 Basic Concepts 3 2 Special Concepts 19 3 Congruences 35 4 Planar Semimodular Lattices 47 Bibliography 325 ix Contents Table of Notation xiii Picture Gallery xvii Preface to the Second Edition xix Introduction xxi I A Brief Introduction to Lattices 1 1 Basic Concepts 3 1.1 Ordering3 1.1.1 Ordered sets3 1.1.2 Diagrams5 1.1.3 Constructions of ordered sets5 1.1.4 Partitions7 1.2 Lattices and semilattices9 1.2.1 Lattices9 1.2.2 Semilattices and closure systems 10 1.3 Some algebraic concepts 12 1.3.1 Homomorphisms 12 1.3.2 Sublattices 13 1.3.3 Congruences 15 2 Special Concepts 19 2.1 Elements and lattices 19 2.2 Direct and subdirect products 20 2.3 Terms and identities 23 2.4 Gluing 28 xi xii Contents 2.5 Modular and distributive lattices 30 2.5.1 The characterization theorems 30 2.5.2 Finite distributive lattices 31 2.5.3 Finite modular lattices 32 3 Congruences 35 3.1 Congruence spreading 35 3.2 Finite lattices and prime intervals 38 3.3 Congruence-preserving extensions and variants 41 4 Planar Semimodular Lattices 47 4.1 Basic concepts 47 4.2 SPS lattices 48 4.3 Forks 49 4.4
    [Show full text]
  • 8. Distributive Lattices Every Dog Must Have His Day. in This Chapter and The
    8. Distributive Lattices Every dog must have his day. In this chapter and the next we will look at the two most important lattice varieties: distributive and modular lattices. Let us set the context for our study of distributive lattices by considering varieties generated by a single finite lattice. A variety V is said to be locally finite if every finitely generated lattice in V is finite. Equivalently, V is locally finite if the relatively free lattice FV(n) is finite for every integer n> 0. Theorem 8.1. If L is a finite lattice and V = HSP(L), then n |L| |FV(n)| ≤ |L| . Hence HSP(L) is locally finite. Proof. If K is any collection of lattices and V = HSP(K), then FV(X) =∼ FL(X)/θ where θ is the intersection of all homomorphism kernels ker f such that f : FL(X) → L for some L∈ K. (This is the technical way of saying that FL(X)/θ satisfies exactly the equations that hold in every member of K.) When K consists of a single finite lattice {L} and |X| = n, then there are |L|n distinct mappings of X into L, and n n 1 hence |L| distinct homomorphisms fi : FL(X) → L (1 ≤ i ≤ |L| ). The range of each fi is a sublattice of L. Hence FV(X) =∼ FL(X)/θ with θ = ker fi means n that FV(X) is a subdirect product of |L| sublattices of L, and so a sublattice of n T n |L| |L| the direct product 1≤i≤|L|n L = L , making its cardinality at most |L| .
    [Show full text]
  • Subdirect Products of Rings and Distributive Lattices
    Proceedings of the Edinburgh Mathematical Society (1982) 25, 155-171 !Q SUBDIRECT PRODUCTS OF RINGS AND DISTRIBUTIVE LATTICES by HANS-J. BANDELT AND MARIO PETRICH* (Received 20th October 1980) 1. Introduction and summary Rings and distributive lattices can both be considered as semirings with commutative regular addition. Within this framework we can consider subdirect products of rings and distributive lattices. We may also require that the semirings with these restrictions are regarded as algebras with two binary operations and the unary operation of additive inversion (within the additive subgroup of the semiring). We can also consider distributive lattices with the two binary operations and the identity mapping as the unary operation. This makes it possible to speak of the join of ring varieties and distributive lattices. We restrict the ring varieties in order that their join with distributive lattices consist only of subdirect products. In certain cases these subdirect products can be obtained via a general construction of semirings by means of rings and distributive lattices. In Section 2 we prove basic results concerning semirings which are lattices of rings. For a semiring of the type under study, i.e. with commutative regular addition, we define two congruences, one of which gives a quotient which is a distributive lattice and the other one gives a ring. The discussion is focused on these congruences and the additive structure mappings arising from the commutative regular addition. Section 3 starts with a consideration of lattices of rings whose characteristics divide a fixed positive integer. This is followed by characterisations of lattices of rings belonging to a fixed variety of arithmetical rings.
    [Show full text]
  • Some Results on Quotients of Triangle Groups
    BULL. AUSTRAL. MATH. SOC. 20F05, 20B30 VOL. 29(1984), 73-90. SOME RESULTS ON QUOTIENTS OF TRIANGLE GROUPS MARSTON D.E. CONDER Given positive integers k,l,m, the (k,l,m) triangle group has presentation A(k,l,m) =< X,Y,Z \ Xk=Yl= Z™ = XYZ = 1> . This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed. 1. Introduction Suppose G is a group which can be generated by two elements, say x and y, such that x has order k and y has order I and their product xy has order m. Quite naturally we call (x,j/) a (k,l,m)- generating pair for G. In such a case, the group G must be a quotient of the {k,l,m) triangle group A(k,l,m), that is, the abstract group with presentation i . _ Y v 7 I "Y" Y '/ 'flit) — ^ A , -I , LJ I A — A — LJ Received 8 February 1984. The author wishes to warmly thank Graham Higman and Tom Tucker, for their inspiration and guidance, also Dave Teague for his assistance with implementation of the Schreier-Sims algorithm, and the University of Otago (Dunedin, New Zealand) for financial support in the form of a post-doctoral fellowship during 1981. Copyright Clearance Centre, Inc.
    [Show full text]