On uncountable hypersimple unidimensional theories

We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable

A dichotomy for DD-rank 1 types in simple theories

We prove a dichotomy for $D$-rank 1 types in simple theories that generalizes Buechler's dichotomy for $D$-rank 1 minimal types in stable theories: every $D$-rank 1 type is either 1-based or part of its algebraic closure, defined by a single formula, almost contains a non-algebraic formula that belongs to a non-forking extension of the type. In addition we prove that a densely 1-based type of $D$-rank 1 is 1-based. We also observe that for a hypersimple unidimensional theory the existence of a non-algebraic stable type implies stability (and thus superstability).Comment: Minor correction made in the statement and proof of Corollary 3.2

Definability and continuity of the SU-rank in unidimensional supersimple theories

We prove, in particular, that in a supersimple unidimensional theory the $SU$-rank is continuous and the $D$-rank is definable

On the forking topology of a reduct of a simple theory

Let $T$ be simple and $T^-$ a reduct of $T$. For variables $x$, we call an $\emptyset$-invariant set $\Gamma(x)$ of ${{\cal C}}$ with the property that for every formula $\phi^-(x,y)\in L^-$: for every $a$, $\phi^-(x,a)$ $L^-$-forks over $\emptyset$ iff $\Gamma(x)\wedge \phi^-(x,a)$ $L$-forks over $\emptyset$, a \em universal transducer\em. We show that there is a greatest universal transducer $\tilde\Gamma_x$ (for any $x$) and it is type-definable. In particular, the forking topology on $S_y(T)$ refines the forking topology on $S_y(T^-)$. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that $\tilde\Gamma_x$ is the unique universal transducer that is $L^-$-type-definable with parameters. In the case where $T^-$ is a theory with the wnfcp (the weak nfcp) and $T$ is the theory of its lovely pairs we show $\tilde\Gamma_x=(x=x)$ and give a more precise description of all its universal transducers in case $T^-$ has the nfcp.Comment: Some examples adde