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.
Abstract: A partially ordered set is said to be directed if every finite subset has an upper bound. In the talk we will introduce the notion of Tukey reducibility between directed sets, and present some basic facts about it. colorings in the free algebra on countably many generators.
Abstract: In this talk we show that nontrivial varieties of algebras enjoy various dual Ramsey properties. The search for dual Ramsey statements has been an important research direction in the past 50 years not only because dual Ramsey results are relatively rare in comparison to the vast number of ``direct'' Ramsey results, but also because they require intricate proof strategies and are usually more powerful than their ``direct'' analogues. It turns out that classes of algebras are a gold mine of dual Ramsey results. In order to prove various dual Ramsey statements for classes of algebras we develop a completely new set of strategies that rely on the fact that right adjoints preserve the Ramsey property while left adjoints preserve the dual Ramsey property. We then consider varieties (that is, equationally defined classes) of algebras as Eilenberg-Moore categories for a monad and show that for every nontrivial variety of algebras all the finite algebras have small dual Ramsey degrees, and that every finite algebra has a finite dual big Ramsey degree with respect to Borel
Abstract: Generalizing the classical results of F. P. Ramsey from the late 1920's, the structural Ramsey theory originated at the beginning of 1970's. In this talk we give a brief overview of fundamental results of structural Ramsey theory, explain the problems that the theory has encountered in its 50 year long history, and introduce the modern apparatus of Ramsey degrees (small and big). In the talk we focus on the categorical reinterpretation of basic notions of Ramsey theory.