# Balcar's theorem on supports

Commentationes Mathematicae Universitatis Carolinae (2018)

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

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topBukovský, 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 -

## References

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