The problem with which this study was concerned was to seek mathematics attitude changes, mathematics self-concept changes, and compare comprehension of concepts in college students enrolled in freshman algebra, when introduced to basic algebraic properties by way of algebra of sets as opposed to an axiomatic introduction.

We study a family of three-dimensional Lie algebras that depend on a continuous parameter. We introduce certain quivers and prove that idempotented versions of the enveloping algebras of the Lie algebras are isomorphic to the path algebras of these quivers modulo certain ideals in the case that the free parameter is rational and non-rational, respectively. We then show how the representation theory of the introduced quivers can be related to the representation theory of quivers of affine type A, and use this relationship to study representations of the family of Lie algebras of interest. In particular, though it is known that this particular family of Lie algebras consists of algebras of wild representation type, we show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations that are of finite or tame representation type.

Vollständig reguläre Halbgruppen weisen eine stark regelmäßige Struktur auf, die verschiedenste Zerlegungsmöglichkeiten gestatten. Ziel dieser Dissertation ist es, diese strukturelle Regelmäßigkeit auf Halbringe zu übertragen und die gewonnenen Algebren zu untersuchen. Mehrere Charakterisierungen werden herausgearbeitet, aufgrund derer es sich herausstellt, dass die Klasse aller vollständig regulären Halbringe eine Varietät bilden, deren Untervarietäten in der Folge untersucht werden. Zentrale Bedeutung haben dabei vollständig einfache Halbringe, deren Analyse einen der Schwerpunkte der Arbeit darstellt. Es zeigt sich, dass diese Bausteine vollständig regulärer Halbringe untereinander eine feste Struktur besitzen, selber aber auch als Zusammensetzung von isomorphen Halbringen aufgefasst werden können. Außerdem werden orthodoxe Halbringe, also Halbringe, deren idempotente Elemente einen Unterhalbring bilden, betrachtet. Zunächst wird dabei wieder auf mehrere Teilklassen eingegangen, bevor abschließend für beliebige vollständig reguläre Halbringe eine Beschreibung der kleinsten Kongruenz angegeben wird, deren Faktorhalbring orthodox ist.

Studiens syfte är att testa, utveckla och jämföra elevers förmåga att lösa förstagradsekvationer samt förenkla algebraiska uttryck där den obekanta variabeln är av, för eleverna, familjär natur, x alternativt y, respektive en godtycklig symbol, till exempel λ eller σ. Deltagarna i studien har gjort ett inledande oförberett test varefter de under höstterminen i årskurs 1 har fått undervisning i ämnet under 32 timmar. Vid terminens slut har eleverna genomfört ett avslutande och likvärdigt test. Totalt har 87 elever på gymnasieskolans teknikprogram deltagit i studien. Materialet i rapporten kommer från elever som började på Ållebergsgymnasiet i Falköping 2011 och 2012. Här visas med 99 procentig och 72 procentig säkerhet att lösning av förstagradsekvationer och förenkling av algebraiska uttryck upplevs som svårare när de genomförs med en generell symbol, som λ eller σ, jämfört med en känd dito, som x, då elever kommer direkt från grundskolan. Det visas även med 85 procentig och 75 procentig säkerhet att lösningsfrekvensen för uppgifterna är högre då symbolen x används. Efter 32 timmars extraundervisning i symbolhanterande algebra, där de introducerats för det grekiska alfabetet, så gick lösningsfrekvenserna på likartade uppgifter upp med 30 – 200 % och hamnade i området 0,85 – 0,95. Skillnaden i upplevd svårighetsgrad och lösningsfrekvens försvann inom ekvationslösningen men kvarstod för förenklingsuppgifter. Studiens resultat visar på en linjär korrelation mellan resultat på det inledande diagnostiska provet och betyg från grundskolan. Elever med betyget G presterar med mer än 90 procentig säkerhet sämre än elever med betyget VG eller MVG. En linjär korrelation mellan lösningsfrekvens och upplevd svårighetsgrad kunde befästas vid såväl det inledande som avslutande testet. Resultaten har jämförts med resultat från en liknande studie som gjordes kring 2000. Vi fann att eleverna i denna studie högst troligt sämre förberedda från grundskolan då de börjar gymnasiet men efter en termin är de minst lika bra eller bättre.

In this thesis we look at hom-associative algebras (which turn out to be exactly the G1-hom-associative algebras), by, in two and three dimensions, trying to find the structure constants for which an algebra becomes hom-associative when the homomorphism 𝛼 is defined as different matrix units. These algebras are also hom-Lie admissible (or G6-hom-associative, which turn out to be the same thing) with a commutator, so we also find the commutator for each of these hom-Lie admissible algebras. We end up finding every hom-associative and hom-Lie algebra for 𝛼 defined as each 2×2 matrix unit in two dimensions, each 3×3 matrix unit in three dimensions when the problem is mapped to one dimension, for three 3×3 matrix units in three dimensions when the problem is mapped to two dimensions (but with the commutators not having been calculated), and only a few hom-associative algebras and hom-Lie algebras for one 3×3 matrix unit in the full three dimensions. We also compare the results for the different values of 𝛼, and find that in 𝑛 dimensions it is possible to find the values of the structure constants for all 𝑛2 different 𝛼:s simply by finding all of the solutions for 𝑛 different 𝛼:s (chosen in a specific way) and then permutating all of the indices.

This bachelor thesis is an introduction to the exterior algebra, interior algebra, regressive algebra, and the (Hodge) star algebra. Together these algebras make up the Grassmann-Cayley algebra endowed with a complement operation giving arise to geometric interpretations with Boolean parallels. Examples from vector calculus, projective geometry and measures are shown.

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The topic of this thesis is the study of moduli spaces of zero-dimensional geometricobjects. The thesis consists of three articles each focusing on a particular moduli space.The first article concerns the Hilbert scheme Hilb(X). This moduli space parametrizesclosed subschemes of a fixed ambient scheme X. It has been known implicitly for sometime that the Hilbert scheme does not behave well when the scheme X is not separated.The article shows that the separation hypothesis is necessary in the sense thatthe component Hilb1(X) of Hilb(X) parametrizing subschemes of dimension zero andlength 1 does not exist if X is not separated.Article number two deals with the Chow scheme Chow 0,n(X) parametrizing zerodimensionaleffective cycles of length n on the given scheme X. There is a relatedconstruction, the Symmetric product Symn(X), defined as the quotient of the n-foldproduct X ×. . .×X of X by the natural action of the symmetric group Sn permutingthe factors. There is a canonical map Symn(X) " Chow0,n(X) that, set-theoretically,maps a tuple (x1, . . . , xn) to the cycle!nk=1 xk. In many cases this canonical map is anisomorphism. We explore in this paper some examples where it is not an isomorphism.This will also lead to some results concerning the question whether the symmetricproduct commutes with base change.The third article is related to the Fulton-MacPherson compactification of the configurationspace of points. Here we begin by considering the configuration space F(X, n)parametrizing n-tuples of distinct ordered points on a smooth scheme X. The schemeF(X, n) has a compactification X[n] which is obtained from the product Xn by a sequenceof blowups. Thus X[n] is itself not defined as a moduli space, but the pointson the boundary of X[n] may be interpreted as geometric objects called stable degenerations.It is then natural to ask if X[n] can be defined as a moduli space of stabledegenerations instead of as a blowup. In the third article we begin work towards ananswer to this question in the case where X = P2. We define a very general modulistack Xpv2 parametrizing projective schemes whose structure sheaf has vanishing secondcohomology. We then use Artin’s criteria to show that this stack is algebraic. Onemay define a stack SDX,n of stable degenerations of X and the goal is then to provealgebraicity of the stack SDX,n by using Xpv2. / QC 20100729

This dissertation deals with the classification of varieties according to their Mal'cev properties. In general the so called Mal'cev-type theorems illustrate an interplay between first order properties of a given class of algebras and the lattice properties of the congruence lattices of algebras of the considered class. CHAPTER 1. A survey of some notational conventions, relevant definitions and auxiliary results is presented. Several examples of less frequently used algebras are given together with the important properties of some of them. The term algebra T(X) and useful results concerning 'term' operations are established. A K-reflection is defined and a connection between a K-reflection of an algebra and whether a class K satisfies an identity of the algebra is established. CHAPTER 2. The Mal'cev-type theorems are presented in complete detail for varieties which are congruence permutable, congruence distributive, arithmetical, congruence modular and congruence regular. Several examples of varieties which exhibit these properties are presented together with the necessary verifications. CHAPTER 3. A general scheme of algorithmic character for some Mal'cev conditions is presented. R. Wille (1970) and A. F. Pixley (1972) provided algorithms for the classification of varieties which exhibit strong Mal'cev properties. This chapter is largely devoted to a modification of the Wille-Pixley schemes. It must be noted that this modification is quite different from all such published schemes. The results are the same as in Wille's scheme but slightly less general than in Pixley's. The text presented here, however is much simpler. As an example, the scheme is used to confirm Mal'cev's original theorem on congruence permutable varieties. Finally, the so-called Chinese var£ety is defined and Mal'cev conditions are established for such a variety of algebras . CHAPTER 4. A comprehensive survey of literature concerning Mal'cev conditions is given in this chapter. / Thesis (M.Sc.)-University of Natal, Durban, 1991.