What's so special about \((\mathbb{Z}/4\mathbb{Z})^ \omega\)?
It is known that any totally categorical structure is built up from a strictly minimal set as a sequence of covers. As the strictly minimal sets in totally categorical structures are classified, to classify totally categorical structures one needs to understand covers. A structure \(M\) is said to be a cover of its 0-definable subset \(N\) if (1) every definable relation on \(N\) is definable with parameters in \(N\) and (2) there are a 0-definable surjection \(f: M\backslash N\to N\) and a definable family \(\{G_ a: a\in N\}\) of groups living in \(N^{eq}\) which act regularly on the fibers \(f^{-1}(a)\) in a definable way. For any structure \(N\) with a definable family \(\{G_ a: a\in N\}\) of groups living in \(N^{eq}\), every cover is a certain expansion of the so-called principal cover \(M_ 0\) which is obtained from \(N\) by adding for every \(a\in A\) a new set \(F_ a\) with a regular action of \(G_ a\) on it and has no other structure. The cover problem is to determine all covering expansions of \(M_ 0\) which induce a new structure on \(N\). The problem turned out to be difficult even in special simple cases. The authors consider the case when \(N\) is the set of nonzero elements of \(\mathcal V\), a countably dimensional vector space over \(\mathbb{F}_ 2\). They solve the cover problem in two special cases: all the groups \(G_ a\) are \((\mathbb{F}_ 2,+)\) (finite type) and each \(G_ a\) is \({\mathcal V}/\mathbb{F}_{2a}\) (affine type). It turns out that all possible covers in the finite case are in some sense `trivial'; in the affine case there are only two `nontrivial' covers which come from the group \((\mathbb{Z}/4\mathbb{Z})^ \omega\), a favorite example in model theory; therefore the title. For countable totally categorical \(N\), the cover problem can be restated in algebraic terms as follows: determine all closed subgroups \(\mathcal G\) of \(\Aut(M_ 0)\) for which the projection map \({\mathcal G}\to\Aut(M_ 0)\) is surjective. In the special cases the authors solve this algebraic problem using some sophisticated cohomological computations.
scientific article; zbMATH DE number 179620