May 2015

# Szymon Żeberski: Applications of Shoenfield Absoluteness Lemma

Tuesday, June 2, 2015 17:15

Room: D1-215

Speaker:
Szymon Żeberski

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

Tuesday, May 26, 2015 18:45

Room: D1-215

Speaker:
Inna Pozdniakova

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

Tuesday, May 26, 2015 17:15

Room: D1-215

Speaker:
Taras Banakh

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

Tuesday, May 19, 2015 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

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

Tuesday, May 12, 2015 17:15

Room: D1-215

Speaker:
Tomasz Żuchowski

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.