Balcar's theorem on supports

Lev Bukovský

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 4, page 443-449
  • ISSN: 0010-2628

Abstract

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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 $u\subseteq D$, $u\in M$.

How to cite

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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},
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
UR - http://eudml.org/doc/294860
ER -

References

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  1. Balcar B., A theorem on supports in the theory of semisets, Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6. Zbl0281.02060MR0340015
  2. Balcar B., Štěpánek P., Set Theory, Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, 2001 (Czech). MR0911270
  3. Jech T., Set Theory, The Third Millenium Edition, Springer Monographs in Mathematics, Springer, 2003. Zbl1007.03002MR1940513
  4. Vopěnka P., Hájek P., The Theory of Semisets, Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, 1972. Zbl0332.02064MR0444473

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