T-algebras and Efimov's problem

We study the topological properties of minimally generated algebras (as introduced by Koppelberg) and, particularly, the subclass of T-algebras (a notion due to Koszmider) and its connection with Efimov’s problem. We show that the class of T-algebras is a proper subclass of the class of minimally generated Boolean algebras. It is also shown that being the Stone space of a T-algebra is not even finitely productive. We prove that the existence of an Efimov T-algebra implies the existence of a coun- terexample for the Stone-Scarborough problem. We also show that the Stone space of an Efimov T -algebra does not map onto the product (?1 + 1) × (? + 1). We establish the following consistency results. Under CH there exists an Efimov min- imally generated Boolean algebra; there are Efimov T-algebra in the forcing extensions obtained by adding ?2 Cohen or Hechler reals to any model of CH

Note on s0s_0 nonmeasurable unions

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ and under some combinatorial and set theoretical assumptions (cov(s_0)=\c for example), that for any family \ca\subseteq s_0 with \bigcup\ca =X, we can find a some subfamily \ca'\subseteq\ca such that the union \bigcup\ca' is not $s$-measurable. We have shown a consistency of the cov(s_0)=\omega_1<\c and existence a partition of the size $\omega_1$ \ca\in [s_0]^{\omega} of the real line \bbr, such that there exists a subfamily \ca'\subseteq\ca for which \bigcup\ca' is $s$-nonmeasurable. We also showed that it is relatively consistent with ZFC theory that \omega_1<\c and existence of m.a.d. family \ca such that \bigcup\ca is $s$-nonmeasurable in Cantor space $2^\omega$ or Baire space $\omega^\omega$. The consistency of $a<cov(s_0)$ and $cov(s_0)<a$ is proved also.Comment: 12 page

Matrix iterations and Cichon's diagram

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon's diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795-810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon's diagram.Comment: 14 pages, 2 figures, article in press for the journal Archive for Mathematical Logi

Supremum vs. maximum: λ-sets

AbstractWe show that, relative to the existence of an inaccessible cardinal, it is consistent that there is no λ-set of maximal size and that in the absence of inaccessible cardinals there is a λ-set of maximal size