Dp-minimality: basic facts and examples

A New Dp-Minimal Expansion of the Integers

“…By multiplicativity of the valuations we may assume that for all formulas of forms ( 5) or ( 6), either n i,1 = n i,2 , n i,1 = 0 or n i,2 = 0. Therefore, by Lemma 2.4, we may assume that every formula of form ( 5) or ( 6) is equivalent to a formula of form (7) or (8). By Lemma 2.3, the conjunction of all the formulas of the forms (3) or ( 4) is equivalent to a formula of the form…”

“…It is enough to find b ′ ∈ M 2 which satisfies this disjunct, along with all the formulas of other forms. Note that v pα (x − s α ) ≥ k α is of form (7), so altogether we want to find b ′ ∈ M 2 which satisfies a conjunction of formulas of the forms: 4 The negation of a formula of form (5)…”

“…Similarly for the negation of a formula of form (6). Also, (7) and ( 8) are in essence special cases of ( 5) or ( 6), but they are required because in A the valuation may be not surjective.…”

“…It is a reduct of the structure (Q p , +, −, •, 0, 1, | p ), which is dp-minimal (see [7,Theorem 6.6]), and therefore is also dp-minimal. Note that Z − is a substructure of…”