Abstract:In {\it A theorem on supports in the theory of semisets} [Comment. Math. Univ. Carolinae {\bf 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 $u\subseteq D$, $u\in M$.
Keywords: inner model; support; generic filter
DOI: DOI 10.14712/1213-7243.2015.266
AMS Subject Classification: 03E40