Models of arithmetic and rich theories

In the present thesis we study the domain of Peano products (in a given model of the Presburger arithmetic (Pr)) as a potentially possible base for a construction of models of the Peano arithmetic (PA). This issue is a special case of the presentation problem which is closely connected to the concept of rich theories. We are especially concerned with one of the basic questions about Peano products domain, i.e. if there exist a pair of Peano products (· , ) such that these products coincide in some slice a: (x)(a · x = a x) and are different below a: (c, d < a)(c · d 6= c d). We reduce this problem to the question if the eliminating set of formulas of the linear arithmetic (LA) is a subset of the set of all existential formulas. We do not solve this problem completely, we only prove that all formulas (x)(z1, z2) , where is a conjunction of equations of terms, are equivalent to existential formulas. We also suggest that the quantifier elimination in the linear arithmetic is considerably more difficult than the elimination in Pr or in the module theory and that it is connected to the problem of description of finitely generated submonoids of Z. We introduce concepts (regular set, standard rationality, saw,... ) and methods which, as we believe, will be essential for an eventual solution of the problem