Chapter 2 Elementary theories and their constructive models

Publisher Summary This chapter presents the algorithmic complexity of relations defined on algebraic structures and discusses the algorithmic properties of various structures by studying their numberings and considering the algorithmic complexity of the set of numbers of elements appearing in the relations under consideration. Models equipped with numberings in which these relations (and elementary properties) are decidable are called “constructive” (strongly constructive), while the models themselves are referred to as “constructivizable” (strongly constructivizable). There are two equivalent approaches to the study of algorithmic properties of algebraic structures. The first approach deals with abstract structures of arbitrary elements and the second approach admits only those structures whose universes consist of natural numbers. The first approach leads to constructive and strongly constructive structures and the second approach leads to recursive and decidable structures. The chapter highlights the existence problem for decidable models with given model-theoretic properties. A study of decidable Boolean algebras, with particular focus on the existence of strongly constructive elementary extensions, is discussed in the chapter.

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