On thin-complete ideals of subsets of groups

Details On thin-complete ideals of subsets of groups On thin-complete ideals of subsets of groups

2010-11-11

arxiv.org/abs/1011.2585v2

Ukr. Math. J. 63 (2011) 741-754

Mathematics

Group Theory

10 pages

Scientific paper

Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $\tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xA\cap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $\tau^*(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.

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