Lattice effect algebras densely embeddable into complete ones

Zdena Riečanová

Kybernetika (2011)

  • Volume: 47, Issue: 1, page 100-109
  • ISSN: 0023-5954

Abstract

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An effect algebraic partial binary operation ø p l u s defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E ^ of E there exists an effect algebraic partial binary operation ⊕ ^ then ⊕ ^ need not be an extension of ⊕ . Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ ^ existing on E ^ is an extension of ⊕ defined on E . Further we show that such ⊕ ^ extending ⊕ exists at most one.

How to cite

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Riečanová, Zdena. "Lattice effect algebras densely embeddable into complete ones." Kybernetika 47.1 (2011): 100-109. <http://eudml.org/doc/197023>.

@article{Riečanová2011,
abstract = {An effect algebraic partial binary operation $øplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat\{E\}$ of $E$ there exists an effect algebraic partial binary operation $\widehat\{\oplus \}$ then $\widehat\{\oplus \}$ need not be an extension of $\{\oplus \}$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat\{\oplus \}$ existing on $\widehat\{E\}$ is an extension of $\{\oplus \}$ defined on $E$. Further we show that such $\widehat\{\oplus \}$ extending $\{\oplus \}$ exists at most one.},
author = {Riečanová, Zdena},
journal = {Kybernetika},
keywords = {non-classical logics; orthomodular lattices; effect algebras; $MV$-algebras; MacNeille completions; orthomodular lattice; effect algebra; MV-algebra; MacNeille completion},
language = {eng},
number = {1},
pages = {100-109},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Lattice effect algebras densely embeddable into complete ones},
url = {http://eudml.org/doc/197023},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Riečanová, Zdena
TI - Lattice effect algebras densely embeddable into complete ones
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 1
SP - 100
EP - 109
AB - An effect algebraic partial binary operation $øplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat{E}$ of $E$ there exists an effect algebraic partial binary operation $\widehat{\oplus }$ then $\widehat{\oplus }$ need not be an extension of ${\oplus }$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat{\oplus }$ existing on $\widehat{E}$ is an extension of ${\oplus }$ defined on $E$. Further we show that such $\widehat{\oplus }$ extending ${\oplus }$ exists at most one.
LA - eng
KW - non-classical logics; orthomodular lattices; effect algebras; $MV$-algebras; MacNeille completions; orthomodular lattice; effect algebra; MV-algebra; MacNeille completion
UR - http://eudml.org/doc/197023
ER -

References

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