Taras Banakh

Shelah groups and their applications in the theory of categorically closed semigroups

In the mini-course we shall discuss recent results related to Shelah groups and their topological properties. A group G is n-Shelah if A^n=G for every subset A\subseteq G of cardinality |A|=|G|. A group is Shelah if it is n-Shelah for some natural number n.  By a result of Protasov, every countable Shelah group is finite, and by a result of Yves de Cornullier, every 3-Shelah group is finite. The first example of an infinite Shelah group was constructed by Shelah in 1980 under CH. His group is 6640-Shelah. We shall explain how to lower the constant 6640 in Shelah’s construction to 36, and mention a recent result of Poor and Rinot who constructed a 10120-Shelah group in ZFC.
Also we shall discuss topological and categorial properties of Shelah groups. The original Shelah group was the first CH-example of non-topologizable group. Later ZFC-examples of (countable) nontopologizable groups were constructed by Hesse (and Olshanskii). We shall explain how to construct a Shelah group G which is absolutely T_1S-closed (which means that for every homomorphism h:G\to Y to a T_1 topological semigroup Y, the image h[G] is closed in Y) but not polybounded (= it cannot be covered by finitely many sets of the form p^{-1}(1) where p(x)=a_0xa_1\dots xa_n is a semigroup polynomial on G). By a result of Banakh and Bardyla, a (countable) group is absolutely T_1S-closed if (and only if) it is polybounded.
If time allows, we shall also discuss various characterizations of (absolutely) T_1S-closed groups and semigroups, elaborated in the joint papers of Banakh and Bardyla.