Quantifier Elimination for Lexicographic Products of Ordered Abelian Groups

Abstract

Let L ag ¼ fþ; À; 0g be the language of the abelian groups, L an expansion of L ag ð<Þ by relations and constants, and L mod ¼ L ag U f1 n g nb2 where each 1 n is defined as follows: x 1 n y if and only if n j x À y. Let H be a structure for L such that H j L ag ð<Þ is a totally ordered abelian group and K a totally ordered abelian group. We consider a product interpretation of H  K with a new predicate I for f0g  K defined by N. Suzuki [9].

Fractional parts of dense additive subgroups of real numbers

“…Item (d) follows from axioms (9), (10). Item (e) follows from axioms (7), ( 9), (10) In section 4, we will show that T is NIP. So, we know that H has a smallest typedefinable subgroup of bounded index [19].…”

“…Item (a) : assume that x < −x, −y < y and y ≤ x; so −x ≤ −y by axiom (4), which implies that x < −x ≤ −y < y, a contradiction; therefore (x < −x & − y < y) → x < y. Item (b) follows from axiom (10). Item (c) follows from axioms (6) I , (6) II .…”

“…Item (c) follows from axioms (6) I , (6) II . Item (d) follows from axioms (9), (10). Item (e) follows from axioms (7), (9), (10).…”

“…Definition 2.4. Let T 0 be the L ρ -theory consisting of the following axioms (1) − (5) and ( 7)− (10). Let T I (respectively T II ) be the theory T 0 together with axiom (6) I (respectively axiom (6) II ).…”

La logique des parties fractionnaires de nombres réels

“…Dans le lemme suivant, l'argument clé avec les projections provenant du lemme 3.4, s'apparente à [4].…”