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National Repository of Grey Literature 2 records found  Search took 0.01 seconds. 
Almost disjoint refinement
 Dohnal, Garik ; Simon, Petr (advisor) ; Hušek, Miroslav (referee)
Construction of completely separable MAD family under the assumption $\mathfrak{s}\leq \mathfrak{a}$ and its relationship with almost disjoint refinement of systems of subsets of $\omega$ on the one side and with topological properties of $\omega^{*}$ on the other. It is shown that the existence of almost disjoint refinement for complements of dense ideals of subsets of $\omega$ is equivalent with the assumption that every nowhere dense set in $\omega^{*}$ is $2^{\omega}$-set. The existence of completely separable MAD family implies these two assumptions. Its construction is proceeded by means of combinatorics properties of systems of sets defined on $\omega$.
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Definable classes of modules and deconstruction of cotorsion pairs
 Dohnal, Garik ; Šaroch, Jan (advisor) ; Šťovíček, Jan (referee)
The goal of this work was to prove the fact, that definable closure of any subclass of cotorsion modules closed under direct sums consists of $\Sigma$-cotorsion modules. The only known proof uses substantially the calculus of derived category, in this work we tried to prove the same, but only by means of a given category of all right $R$-modules and set-theoretic properties of partial orders indexing direct systems of $R$-modules. The main results of this work are proved under additional assumptions on the ring $R$, in particular $\vert R\vert\leq\aleph_{\omega}$ or $\text{dim}(R)<\aleph_{\omega}$. Attempts to give s proof in the same general situation, where the fact is known to hold, was not successful. Powered by TCPDF (www.tcpdf.org)
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