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Register a new userFractional Parts of Dense Additive Subgroups of Real Numbers
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Given a dense additive subgroup $G$ of $mathbb R$ containing $mathbb Z$, we consider its intersection $mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a standard part and two ordered semigroups of infinitely small and infinitely large elements.
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We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupinski]: if $R$ is unital, then $(bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A = bar R^{00}_A$, where $bar R succ R$ is a sufficiently saturated elementary extension of $R$, and $(bar R,+)^{00}_A$ [resp. $bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $bar R$. This implies that $bar R^{00}_A=bar R^{000}_A$, where $bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A = bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(bar R cup {1}) cdot (bar R,+)^{00}_A$ and study related aspects, showing optimality of some of our main results and answering some natural questions.
DAquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $omega_1$-like models of PA.
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.
This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature S, a class ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X ^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromovs proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
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