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Author McKenzie, Ralph. author Structure of Decidable Locally Finite Varieties [electronic resource] / by Ralph McKenzie, Matthew Valeriote Boston, MA : Birkhรคuser Boston, 1989 http://dx.doi.org/10.1007/978-1-4612-4552-0 VIII, 216 p. online resource

SUMMARY

A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propoยญ sitions of arithmetic. During the 1930s, in the work of such mathematiยญ cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church proยญ 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 (Szmielẽ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The deยญ 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

0 Preliminaries -- 1 Preview: The three sub varieties -- I: Structured Varieties -- 2: a property of the center -- 3: Centerless algebras -- 4: The discriminator subvariety -- 5: The Abelian subvariety -- 6: Transfer principles -- Summary of Part I -- II: Structured Abelian Varieties -- 7: Strongly solvable varieties -- 8: More transfer principles -- 9: Consequences of the transfer principles -- 10: Three interpretations -- 11: From strongly Abelian to essentially unary varieties -- 12: The unary case -- III: The Decomposition -- 13: The decomposition theorem -- 14: Conclusion -- Notation

Mathematics Algebra Mathematics Algebra General Algebraic Systems

Location

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