Errata
- The "group-like" case in Zilber's trichotomy
theorem and
conjecture was misstated.
If one wishes to think of these groups as akin to vector spaces over
a division ring, then one must speak of quasi-endomorphisms.
To fix ideas, we restrict to the case of strongly minimal groups.
A quasi-endomorphism of the strongly minimal group G
is a strongly minimal subgroup H < G2
of the square of G whose projection onto the first
coordinate is all of G. The set of quasi-endomorphisms of
G, Q(G),
forms a division ring with the operations:
- H + K := {(x,y+z) | (x,y) ∈ H & (x,z) ∈ K}
- H • K := {(x,z) | (∃ y) (x,y) ∈ K &
(y,z) ∈ H }0
where here (…)0 means to take the
connected component, the smallest definable subgroup of
finite index.
A strongly minimal group G is locally modular just in case
forking in G is always given by
Q(G)-linear dependence.