We investigate the collections of (elementary) submodels of first order structures ordered by the inclusion and some other natural orderings. These partial orders are observed from the aspect of set theory, their cardinal and order invariants are explored and they are examined as forcing notions as well.
The conditions under which forcing violates certain structures of a given model of set theory, such as ultrafilters, maximal almost disjoint families and inseparable sequences, are investigated.
We study the condensational preorder on the class of all relational structures, induced by bijective homomorphisms (condensations), and the corresponding equivalence relation of bi-condensability. Emphasis is put on investigating the class of structures that have the property Cantor-Schroeder-Bernstein with respect to that preorder, including the structures for which each bijective endomorphism is an automorphism (the so called reversible structures).
The cut-and-choose games on Boolean algebras are examined, searching for equivalent conditions for the existence of winning strategies and for the examples of Boolean algebras on which the games have different outcomes.
The topologies on complete Boolean algebras generated by convergence structures are explored. The relations between the topological properties of the spaces obtained in this way and the algebraic and forcing properties of the corresponding Boolean algebras are examined.
From a starting topology there are several ways to generate a new topology by an ideal on the ground set. We study differences and similarities between those new topologies, depending on properties of the starting topology and the ideal.