The right angle to look at orthogonal sets

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developped. Moreover, a version of Schlichting's Theorem for hyperdefinable families of commensurable subgroups is shown

SHELAH-STRONG TYPE AND ALGEBRAIC CLOSURE OVER A HYPERIMAGINARY (Model theoretic aspects of the notion of independence and dimension)

We characterize Shelah-strong type over a hyperimagianary with the algebraic closure of a hyperimaginary. Also, we present and take a careful look at an example that witnesses acl[eq](ℯ) is not interdefinable with acl(ℯ) where ℯ is a hyperimaginary

On piecewise hyperdefinable groups

The aim of this paper is to generalize and improve two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the stabilizer theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies