Marek Bienias

# Marek Bienias: About universal structures and Fraisse theorem

Tuesday, April 24, 2018 17:15

Room: D1-215

Speaker:
Marek Bienias

Title: About universal structures and Fraisse theorem

Abstract. For a given structure D of language L we can consider age of D, i.e. the family of all finitely generated L-substructures od D. It turns out that age has property (HP) and (JEP). Fraisse theorem let us revers the procedure: if K is nonempty countable family of finitely generated L-structures having properties (HP), (JEP) and (AP), then there exists exactly one (up to isomorphism) L-structure D (so called Fraisse limit) which is countable ultrahomogenous and has age K.
The aim of the talk is to define basic notions from Fraisse theory, proof the main theorem and show some alternative way of looking at the construction of Fraisse limit.

# Marek Bienias: General methods in algebrability

Tuesday, April 14, 2015 17:15

Room: D1-215

Speaker:
Marek Bienias

Title: General methods in algebrability

Abstract: During last 15 years new idea of measuring sets appeared and become popular.

Def.
Let $$\kappa$$ be a cardinal number and let $$\mathcal{L}$$ be a commutative algebra. Assume that $$A\subseteq\mathcal{L}$$. We say that $$A$$ is:
• $$\kappa$$-algegrable if $$A\cup \{0\}$$ contains $$\kappa$$-generated algebra $$B$$;
• strongly $$\kappa$$-algegrable if $$A\cup \{0\}$$ contains $$\kappa$$-generated free algebra $$B$$.

In many recent articles authors studied algebrability of sets naturally appering in mathematical analysis. It seems that required results are the general methods of algebrability which can cover known methods and give new constructions.

We will describe two methods:  independent Bernstein sets and exponential like. They let us prove many results concerning algebrability and strong algebrability of subsets of algebras $$\mathbb{R}^\mathbb{R}$$, $$\mathbb{C}^\mathbb{C}$$, $$\mathbb{R}^\mathbb{N}$$, $$C[0,1]$$, $$\mathcal{l}_\infty$$. Most of presented applications give the best possible result in terms of complication of built algebraic structure and cardinality of set of generators of this structure (in most cases $$\mathfrak c$$ or $$2^{\mathfrak c}$$).