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

Binary simple homogeneous structures are supersimple with finite rank

Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set