Novi Sad group for

Set Theory, Model Theory and General Topology

Department of Mathematics and Informatics

Faculty of Sciences

University of Novi Sad

Trg Dositeja Obradovića 4,

Novi Sad, Serbia

E-mail : settop@dmi.uns.ac.rs

Supported by:

Ministry of Education,

Science and Technological Development

of the Republic of Serbia

Project number 174006

Science Fund of the Republic of Serbia

SEMINAR "FUNDAMENTI MATEMATIKE"

Tuesday, April 5, 2022 - 16:00 CET

Boris Šobot (Univerzitet u Novom Sadu)

**Apstrakt:** Neka je betaN skup ultrafiltera na skupu N prirodnih brojeva,
i elementi skupa N su izjednačeni sa glavnim ultrafilterima. Kvaziuređenje
| na betaN, ekstenzija relacije deljivosti na N, proučavano je u
nekoliko prethodnih radova predavača.
Kongruencija po modulu m (m prirodan broj)
može se proširiti na prirodan način na betaN.
Ispitaćemo, za početak, u kojoj meri se ovo proširenje slaže sa |.
Potom ćemo razmotriti dva načina da se definiše i kongruencija po modulu U,
gde je U neglavni ultrafilter. Drugi od ta dva načina vodi nas do prirodnog
pojačanja relacije deljivosti ultrafiltera |. Razmotrićemo i koliko su ove
nove relacije kongruentnosti saglasne sa relacijama deljivosti.

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.