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Register a new userThorn independence in the field of real numbers with a small multiplicative group
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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.
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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.
We improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $Sigma_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalban. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.
We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.
Danos and Regnier (1989) introduced the par-switching condition for the multiplicative proof-structures and simplified the sequentialization theorem of Girard (1987) by the use of par-switching. Danos and Regner (1989) also generalized the par-switching to a switching for $n$-ary connectives (an $n$-ary switching, in short) and showed that the expansion property which means that any excluded-middle formula has a correct proof-net in the sense of their $n$-ary switching. They added a remark that the sequentialization theorem does not hold with their switching. Their definition of switching for $n$-ary connectives is a natural generalization of the original switching for the binary connectives. However, there are many other possible definitions of switching for $n$-ary connectives. We give an alternative and natural definition of $n$-ary switching, and we remark that the proof of sequentialization theorem by Olivier Laurent with the par-switching works for our $n$-ary switching; hence that the sequentialization theorem holds for our $n$-ary switching. On the other hand, we remark that the expansion property does not hold with our switching anymore. We point out that no definition of $n$-ary switching satisfies both the sequentialization theorem and the expansion property at the same time except for the purely tensor-based (or purely par-based) connectives.
The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent achievements on equipping them with a derivation, which is done by proving that surreal numbers can be seen as transseries and by finding the `simplest structure of H-field, the abstract version of a Hardy field. All the latter notions and their context are also addressed, as well as the universality of the resulting structure for surreal numbers.
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