Balcar's theorem on supports

Bukovský, Lev

Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 4, pp. 443-449.

Résumé

In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1--6] B. Balcar showed that if $\sigma\subseteq D\in M$ is a support, $M$ being an inner model of ZFC, and ${\mathcal P}(D\setminus \sigma)\cap M=r``\sigma$ with $r\in M$, then $r$ determines a preorder "$\preceq$" of $D$ such that $\sigma$ becomes a filter on $(D,\preceq)$ generic over $M$. We show that if the relation $r$ is replaced by a function ${\mathcal P}(D\setminus \sigma)\cap M=f_{-1}(\sigma)$, then there exists an equivalence relation "$\sim$" on $D$ and a partial order on $D/\sim\,$ such that $D/\sim\,$ is a complete Boolean algebra, $\sigma/\sim\,$ is a generic filter and $[f(u)]_{\sim}=-\sum (u/\sim)$ for any $u\subseteq D$, $u\in M$.

Détail

DOI : 10.14712/1213-7243.2015.266

Classification : 03E40
Keywords: inner model; support; generic filter