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Quasigroups, Right Quasigroups, Coverings and Representations Tengku Simonthy Renaldin Fuad Iowa State University

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Quasigroups, Right Quasigroups, Coverings and Representations Tengku Simonthy Renaldin Fuad Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1993 Quasigroups, right quasigroups, coverings and representations Tengku Simonthy Renaldin Fuad Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Fuad, Tengku Simonthy Renaldin, "Quasigroups, right quasigroups, coverings and representations " (1993). Retrospective Theses and Dissertations. 10427. https://lib.dr.iastate.edu/rtd/10427 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 Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ivll 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9321144 Quasigroups, right quasigroups, coverings, and representations Fuad, Tengku Simonthy Renaldin, Ph.D. Iowa State University, 1993 Copyright ©1993 by Fuad, Tengku Simonthy Renaldin. All rights reserved. UMI 300 N. ZeebRd. Ann Arbor, MI 48106 Quasigroups, right quasigroups, coverings, and representations by Tengku Simonthy Renaldin Fuad A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Mathematics Major: Mathematics Approved: Members of the Committee: Signature was redacted for privacy. rge of Major Work Signature was redacted for privacy. Signature was redacted for privacy. Fo^ Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1993 Copyright ©Tengku Simonthy Renaldin Fuad, 1993. All rights reserved. ii TABLE OF CONTENTS INTRODUCTION 1 CHAPTER 1. GROUP MODULES 3 CHAPTER 2. REPRESENTATIONS OF QUASIGROUPS 6 CHAPTER 3. REPRESENTATIONS OF RIGHT QUASIGROUPS 27 CHAPTER 4. MODULES IN A VARIETY OVER A RING 50 SUMMARY 61 REFERENCES 62 ACKNOWLEDGEMENT 64 1 INTRODUCTION The theory of quasigroup modules was laid out clearly in [S2]. The theory has also been described as quasigroup representation theory, and depends heavily on the construction of the universal multiplication group. The theory of universal multiplication group has been studied for some time now, e.g. [P]. In [S2], it has been shown that quasigroup representation theory is equivalent to the representation theory of stabilizers in the universal multii)lication group. To every knot, there has been associated an invariant called the fundamental group. In general, we have Poincare's fundamental group functor from a topological space, A generalization of this fundamental group is the so-called fundamental groupoid [B]. It is a fact that covering spaces of topological spaces can be classified by their fundamental groupoids. Another generalization of fundamental groupoids and cov­ erings in the directed graph case are given in [H]. The strongest invariant of a knot, the so-called knot quandle, was found and studied by [J]. The concept of algebroid and left algebroid modules has been described in [M] and [R]. In [M], an algebroid has been defined as a category equipped with module structure on each hom set such that the composition is bilinear. In [R] an algebroid modules has been defined as a fc-functor from an algebroid to the category of modules. Chapter 1 of this dissertation explains the theory of group modules which is the background of the theory of quasigroup modules. Chapter 2 includes the theory of quasigroup modules with some new results, e.g. interpreting quasigroup mod­ ules as representations of the fundamental groupoid of the Cay ley diagram of the quasigroup in the category of abelian groups. The equivalent coverings will also be described here. Chapter 3 is concerned with the theory of right quasigroup modules. This consists of generalization of the new results obtained in Chapter 2. Chapter 4 is about the theory of right quasigroup modules in a variety over a fixed field. Here we will need the concept of left-algebroid module. A new category will also be 2 defined, the so-called category of ordered modules in a fixed variety. In particular, modules over quandles are studied here. Part of this thesis is based on the talks given by the author at Iowa State Uni­ versity seminars, the Alan Day Conference, and the AMS meeting in San Antonio, Texas in January 1993. 3 CHAPTER 1 GROUP MODULES Representations of a group G are usually considered as abelian groups (M, +) on which G acts, i.e. there is a group morphism G —> rm Aut(M, +). Categorically, this means that there is a functor from G to rm Aut(M, +)• However, there is another approach to looking at modules M, for a group G, This consists of build­ ing the split extension M ] G, together with its projection TT : M ] G —» G. The underlying set of M ] G is M x G, equipped with the group operation. (mi,(7i)(m2,<72) = {m The map tt: M] G —> G; (m,g) gr is a group homomorphism [in fact even onto]. We can think of Tr~^(g) as the fibre of </ G G, isomorphic to M [or in other words, "M located at g"]. A G-module morphism ip: M M' is an abelian group homomorphism such that {mip)g = {rng)if. This induces a group homomorphism Tp\ M]G —> M']G; (m,g) w (m(f,g) such that the following diagram commutes. M]G —^ M']G TT G G On the other hand, a group homomorphism / : M ] G —> M' ] G induces a module homomorphism f : M M' defined by (m/', 1) = (m,l)/. One can then think about the category 0/G of groups over G with the above structure. Let D be the graph ^2 Then the pull back of D is the left limit L L AI /a «1 A2 > ^0 «2 Analogously, the pull back of G' ^ G G" is the group H defined by H = G'XG G" = {{g',g") € G'X G"\g'7R' = g"7R"}. Prom each object in this category &/G, there exists a morphism to 1: G —G, which makes 1 the terminal object of 0/G. Formulating everything in the category Set of sets, an abelian group A is a set A with maps 0: {1} = A^ —* A, A A, and +: A'^ A such that various identities are satisfied. These identities can be written as commuting diagrams in the category Set of sets. Generalizing, define an abelian group A in (5/G to be an object A of (5/G equipped with (5/G-morphisms 0: —> A, — : A —> A, and +; A^ —> A. The diagrams representing abelian group identities, now interpreted as diagrams in 0/G, are required to commute. For example the "abelian" property is the commuting diagram T AXQ A > AXQ A An example of an abelian group in 0/G is M] G G, where Og: G -> M] G; <7 (0,g) M] G M] G; (m,g) 1-^ {-m,g) +: M]GXGM]G->M]G; ((MI, </), (M2, <7)) I-> (MI +M2,G). 5 Hence each G-module M gives an element M] C? —> G oî<Ô/G. Suppose we have an abelian group H G'm (5/G. Clearly TT is an epimorphism in 0, with kernel congruence a. Identify H°' with G via the natural isomorphism. Consider the kernel oî H XQ H H XQ H H, say (ala). Then («lo;) satisfies certain properties: (CO) (a,6)(O!|<X)(C,d) =4" a — b = c—d a" = (a — 6)" = (c — d)" = c". (Cl) For fixed (o, 6) in a, the mapping (a, 6)('^l°) —> a"; (c, rf) i-> c has the two- sided inverse c i—> (c, c — (a — b)). (C2) : (RR) If a" — c", then a—a = a^Oq = C°OQ = c—c. So that (a, A)(Q;|Q!)(C, c). (RS) (a,b)(ala)(c,d) => a — b = c — d b — a = d—c ==^ (6, a)(a|a!)((Z, c) (RT) {a,b)(a\a)(c,d) and (b, e)(a\a){d, f) ==^ a — b = c — d and b — e = d — f =» a — e — c — f =» {a,e)(a\a)(c,f). We also will have an isomorphism from —> H°' = G hy defining /: -> if"; (o,6)(^l») a"" to H ^ G by the mutually inverse 0/G-morphisms j : H «("I"); a H-» (a, and k : &(«!«) -> H] (a, 6)(«l") ^a-b. We will say that the congruence («1a) on a is the centreing congruence by which a centralizes itself.
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