May 2015

# Szymon Żeberski: Applications of Shoenfield Absoluteness Lemma

27/05/15 21:38

Tuesday, June 2, 2015 17:15

*Room:*D1-215*Szymon Żeberski*

Speaker:Speaker:

*Title*: Applications of Shoenfield Absoluteness Lemma*Abstract*. We will recall Shoenfield Absoluteness Lemma about \(\Sigma^1_2\) sentences. We will show applications of this theorem connected to topological and algebraic structure of Polish spaces in publications co-authored by the speaker.# Inna Pozdniakova: On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images

21/05/15 17:01

Tuesday,

**May 26**, 2015**18:45***Room:*D1-215*Inna Pozdniakova*

Speaker:Speaker:

*Title*: On monoids of monotone injective partial selfmaps of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) with co-finite domains and images*Abstract*. The speaker will discuss on the structure of the semigroup \(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})\) of monotone injective partial selfmaps of the set of \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) having co-finite domain and image, where \(L_n\times_{\operatorname{lex}}\mathbb{Z}\) is the lexicographic product of an \(n\)-elements chain and the set of integers with the usual order.# Taras Banakh : Separation axioms on paratopological groups and quasi-uniform spaces

20/05/15 10:19

Tuesday, May 26, 2015 17:15

*Room:*D1-215*Taras Banakh*

Speaker:Speaker:

*Title*: Separation axioms on paratopological groups and quasi-uniform spaces*Abstract*. We shall prove that each regular paratoplogical group is completely regular thus resolving an old problem in the theory of paratopological groups.# Jarosław Swaczyna: Generalized densities of subsets of natural numbers and associated ideals

15/05/15 19:36

Tuesday, May 19, 2015 17:15

I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).

I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.

*Room:*D1-215*Jarosław Swaczyna*

Speaker:Speaker:

*Title*: Generalized densities of subsets of natural numbers and associated ideals*Abstract*. Let \(g: \omega \rightarrow [0, \infty)\). We say that \(A \subset \omega\) has \(g\)-density zero, if \(\lim_{n \rightarrow \infty} \frac{A \cap n}{g(n)} = 0\). It is an easy observation that family of \(g\)-density zero sets is an ideal.I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).

I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.

# Tomasz Żuchowski: Tukey types of orthogonal ideals

08/05/15 08:24

Tuesday, May 12, 2015 17:15

*Room:*D1-215*Tomasz Żuchowski*

Speaker:Speaker:

*Title*: Tukey types of orthogonal ideals*Abstract*. A partial order \(P\) is Tukey reducible to partial order \(Q\) when there exists a function \(f:P\to Q\) such that if \(A\) is a bounded subset of \(Q\) then \(f^{-1}[A]\) is a bounded subset of \(P\). The existence of such reduction is related to some cardinal invariants of considered orders. We will show Tukey reductions between some special ideals of subsets of \(\mathbb{N}\) with the inclusion order and other partial orders.