An algebraic version of the Cantor-Bernstein-Schröder theorem

Hector Freytes

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 3, page 609-621
  • ISSN: 0011-4642

Abstract

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The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to σ -complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for σ -complete orthomodular lattices, Stone algebras, B L -algebras, M V -algebras, pseudo M V -algebras, Łukasiewicz and Post algebras of order n .

How to cite

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Freytes, Hector. "An algebraic version of the Cantor-Bernstein-Schröder theorem." Czechoslovak Mathematical Journal 54.3 (2004): 609-621. <http://eudml.org/doc/30886>.

@article{Freytes2004,
abstract = {The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to $\sigma $-complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for $\sigma $-complete orthomodular lattices, Stone algebras, $BL$-algebras, $MV$-algebras, pseudo $MV$-algebras, Łukasiewicz and Post algebras of order $n$.},
author = {Freytes, Hector},
journal = {Czechoslovak Mathematical Journal},
keywords = {lattices; central elements; factor congruences; varieties; lattices; central elements; factor congruences; varieties},
language = {eng},
number = {3},
pages = {609-621},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An algebraic version of the Cantor-Bernstein-Schröder theorem},
url = {http://eudml.org/doc/30886},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Freytes, Hector
TI - An algebraic version of the Cantor-Bernstein-Schröder theorem
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 609
EP - 621
AB - The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to $\sigma $-complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for $\sigma $-complete orthomodular lattices, Stone algebras, $BL$-algebras, $MV$-algebras, pseudo $MV$-algebras, Łukasiewicz and Post algebras of order $n$.
LA - eng
KW - lattices; central elements; factor congruences; varieties; lattices; central elements; factor congruences; varieties
UR - http://eudml.org/doc/30886
ER -

References

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