Set Theory, Model Theory and Set-Theoretic Topology

POSETS OF ELEMENTARY SUBMODELS: The object of the research will be the collections of elementary submodels of first order structures, ordered by the inclusion and some other natural orderings. These partial orders will be observed from the aspect of set theory, their cardinal and order invariants will be explored, and they will be examined as forcing notions as well. SET-THEORETIC FORCING: 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, will be investigated. GAMES ON BOOLEAN ALGEBRAS: The cut-and-choose games on Boolean algebras will be 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. CONVERGENCE STRUCTURES ON BOOLEAN ALGEBRAS: The topologies on complete Boolean algebras generated by convergence structures will be 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 will be examined. MODEL-THEORETIC FORCING: The relations between both n-finite and n-infinite forcing companions of a given theory will be investigated, with the aim of investigating the theory itself and getting a better image of the role of more complex sentences.