Balcar's theorem on supports

Bukovský, Lev. "Balcar's theorem on supports." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 443-449. <http://eudml.org/doc/294860>.

@article{Bukovský2018,
abstract = {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 $u\subseteq D$, $u\in M$.},
author = {Bukovský, Lev},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {inner model; support; generic filter},
language = {eng},
number = {4},
pages = {443-449},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Balcar's theorem on supports},
url = {http://eudml.org/doc/294860},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Bukovský, Lev
TI - Balcar's theorem on supports
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 443
EP - 449
AB - 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 $u\subseteq D$, $u\in M$.
LA - eng
KW - inner model; support; generic filter
UR - http://eudml.org/doc/294860
ER -