Ebook: Structure of Decidable Locally Finite Varieties
- Tags: Algebra, General Algebraic Systems
- Series: Progress in Mathematics 79
- Year: 1989
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-viii
Front Matter....Pages 1-3
Preliminaries....Pages 5-33
Preview: The three subvarieties....Pages 35-36
Front Matter....Pages 37-37
A property of the center....Pages 39-55
Centerless algebras....Pages 57-64
The discriminator subvariety....Pages 65-71
The Abelian subvariety....Pages 73-74
Transfer principles....Pages 75-89
Back Matter....Pages 91-91
Front Matter....Pages 93-93
Strongly solvable varieties....Pages 95-98
More transfer principles....Pages 99-102
Consequences of the transfer principles....Pages 103-105
Three interpretations....Pages 107-128
From strongly Abelian to essentially unary varieties....Pages 129-148
The unary case....Pages 149-167
Front Matter....Pages 169-169
The decomposition theorem....Pages 171-192
Conclusion....Pages 193-197
Back Matter....Pages 199-215
Content:
Front Matter....Pages i-viii
Front Matter....Pages 1-3
Preliminaries....Pages 5-33
Preview: The three subvarieties....Pages 35-36
Front Matter....Pages 37-37
A property of the center....Pages 39-55
Centerless algebras....Pages 57-64
The discriminator subvariety....Pages 65-71
The Abelian subvariety....Pages 73-74
Transfer principles....Pages 75-89
Back Matter....Pages 91-91
Front Matter....Pages 93-93
Strongly solvable varieties....Pages 95-98
More transfer principles....Pages 99-102
Consequences of the transfer principles....Pages 103-105
Three interpretations....Pages 107-128
From strongly Abelian to essentially unary varieties....Pages 129-148
The unary case....Pages 149-167
Front Matter....Pages 169-169
The decomposition theorem....Pages 171-192
Conclusion....Pages 193-197
Back Matter....Pages 199-215
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