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Keywords:
inner model; support; generic filter
inner model; support; generic filter
Summary:
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 $u\subseteq D$, $u\in M$.
References:
[1] Balcar B.: A theorem on supports in the theory of semisets. Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6. MR 0340015 | Zbl 0281.02060
[2] Balcar B., Štěpánek P.: Set Theory. Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, 2001 (Czech). MR 0911270
[3] Jech T.: Set Theory. The Third Millenium Edition, Springer Monographs in Mathematics, Springer, 2003. MR 1940513 | Zbl 1007.03002
[4] Vopěnka P., Hájek P.: The Theory of Semisets. Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, 1972. MR 0444473 | Zbl 0332.02064
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