Structure of Decidable Locally Finite Varieties

1. Front Matter

   Pages i-viii

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2. Introduction

   1. Front Matter

      Pages 1-3

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   2. Preliminaries

      • Ralph McKenzie, Matthew Valeriote

      Pages 5-33

   3. Preview: The three subvarieties

      • Ralph McKenzie, Matthew Valeriote

      Pages 35-36

3. Structured varieties

   1. Front Matter

      Pages 37-37

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   2. A property of the center

      • Ralph McKenzie, Matthew Valeriote

      Pages 39-55

   3. Centerless algebras

      • Ralph McKenzie, Matthew Valeriote

      Pages 57-64

   4. The discriminator subvariety

      • Ralph McKenzie, Matthew Valeriote

      Pages 65-71

   5. The Abelian subvariety

      • Ralph McKenzie, Matthew Valeriote

      Pages 73-74

   6. Transfer principles

      • Ralph McKenzie, Matthew Valeriote

      Pages 75-89

4. Back Matter

   Pages 91-91

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5. Structured Abelian varieties

   1. Front Matter

      Pages 93-93

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   2. Strongly solvable varieties

      • Ralph McKenzie, Matthew Valeriote

      Pages 95-98

   3. More transfer principles

      • Ralph McKenzie, Matthew Valeriote

      Pages 99-102

   4. Consequences of the transfer principles

      • Ralph McKenzie, Matthew Valeriote

      Pages 103-105

   5. Three interpretations

      • Ralph McKenzie, Matthew Valeriote

      Pages 107-128

   6. From strongly Abelian to essentially unary varieties

      • Ralph McKenzie, Matthew Valeriote

      Pages 129-148

   7. The unary case

      • Ralph McKenzie, Matthew Valeriote

      Pages 149-167

6. The decomposition

   1. Front Matter

      Pages 169-169

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   2. The decomposition theorem

      • Ralph McKenzie, Matthew Valeriote

      Pages 171-192

A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo­ sitions of arithmetic. During the 1930s, in the work of such mathemati­ cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro­ posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de­ termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.

• Abelian group
• Boolean algebra
• Finite
• Mathematica
• algebra
• algorithms
• boundary element method
• decidability
• eXist
• function
• geometry
• language
• set
• system
• theorem

• Department of Mathematics, University of California, Berkeley, USA

  Ralph McKenzie

• Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

  Matthew Valeriote

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