Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 106-108).

Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-18T21:11:32Z No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:18:27Z (GMT) No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T18:22:43Z (GMT) No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) / Made available in DSpace on 2015-07-06T18:22:43Z (GMT). No. of bitstreams: 1 Dissertação-Luiz C C Mendes.pdf: 1127626 bytes, checksum: 3867dce3c1616f07fa4984ef6374d5d5 (MD5) Previous issue date: 2015-02-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aims to introduce students and teachers of basic mathematical education some resolutions of problems in the field of arithmetic which can benefit the teachinglearning process. Initially, it will be addressed the divisibility with their properties and criteria. This is done after a presentation of Euclidean division and its applications in basic education. Moreover, it will be presented a brief theoretical background based on the concept and the operational properties of modular congruence with their residue classes, followed by their applications. Finally, it will be presented a brief history of the numbers in the calendars. / A presente dissertação tem como objetivo principal apresentar a alunos e professores de matemática do ensino básico algumas resoluções de problemas no campo da aritmética que pode beneficiar o processo ensino-aprendizagem. Serão abordados inicialmente a divisibilidade, com suas propriedades e seus critérios, após apresentação da divisão euclidiana e suas aplicações no ensino básico. Além disso, será apresentado um breve embasamento teórico, pautado no conceito e nas propriedades operacionais da congruência modular com suas classes residuais, seguido de suas aplicações. No final, será feito um breve histórico dos números nos calendários.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2016-12-12T15:53:35Z No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-13T19:31:20Z (GMT) No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-13T19:31:20Z (GMT). No. of bitstreams: 2 Dissertação - Débora Danielle Alves Moraes Priebe - 2016.pdf: 1557477 bytes, checksum: 54ff96d96b239797a8305d6ff67e2f12 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-07 / This paper aims to present an educational proposal of some topics of arithmetic, also called Number Theory, for the final grades of elementary school, focusing on solving problems to challenge and entertain students with the range of possibilities arising from properties of Number Theory and develop their thinking skills through interesting problems that will give a new life to the subject . The reader will find in this work topics of divisibility, primes, Greatest Common Divisor, Least Common Multiple, Euclidean Algorithm, congruences, decimal representation, divisibility tests, as well as several examples, challenging problems and also curiosities about the congruence module 9. / Este trabalho tem como objetivo apresentar uma proposta de ensino de alguns tópicos de Aritmética, também denominada de Teoria dos Números, às séries finais do Ensino Fundamental, com foco na resolução de problemas, visando desafiar e fascinar os alunos com a gama de possibilidades oriunda das propriedades da Teoria dos Números e desenvolver sua capacidade de raciocínio através de problemas interessantes que darão uma nova vida ao assunto. O leitor encontrará neste trabalho tópicos de divisibilidade, primos, Máximo Divisor Comum, Mínimo Múltiplo Comum, Algoritmo de Euclides, congruências, representação decimal, testes de divisibilidade, além de diversos exemplos, problemas desafiadores e também curiosidades acerca da congruência módulo 9.