Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

Martin Kalina

Kybernetika (2010)

  • Volume: 46, Issue: 6, page 935-947
  • ISSN: 0023-5954

Abstract

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If element z of a lattice effect algebra ( E , ⊕ , 0 , 1 ) is central, then the interval [ 0 , z ] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕ . It is a known fact that if the set C ( E ) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C ( E ) in E equals to the top element of E , then E is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion E ^ of E which is its extension (i.e. E is densely embeddable into E ^ ) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of E ^ . For example, the existence of a state in E ^ implies the existence of a state in E . In this context, a natural question arises if the MacNeille completion of the center of E (denoted as ℳ 𝒞 ( C ( E ) ) ) is necessarily the same as the center of E ^ , i.e., if ℳ 𝒞 ( C ( E ) ) = C ( E ^ ) is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of C ( E ) and its bifullness in E is not sufficient to guarantee the mentioned equality.

How to cite

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Kalina, Martin. "Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras." Kybernetika 46.6 (2010): 935-947. <http://eudml.org/doc/196610>.

@article{Kalina2010,
abstract = {If element $z$ of a lattice effect algebra $(E,\oplus , \{\mathbf \{0\}\}, \{\mathbf \{1\}\})$ is central, then the interval $[\{\mathbf \{0\}\},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus $. It is a known fact that if the set $C(E)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C(E)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\hat\{E\}$ of $E$ which is its extension (i.e. $E$ is densely embeddable into $\hat\{E\}$) then it is possible to embed $E$ into a direct product of irreducible effect algebras. Thus $E$ inherits some of the properties of $\hat\{E\}$. For example, the existence of a state in $\hat\{E\}$ implies the existence of a state in $E$. In this context, a natural question arises if the MacNeille completion of the center of $E$ (denoted as $\{\mathcal \{M\}\}\{\mathcal \{C\}\}(C(E))$) is necessarily the same as the center of $\hat\{E\}$, i.e., if $\{\mathcal \{M\}\}\{\mathcal \{C\}\}(C(E))=C(\hat\{E\})$ is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of $C(E)$ and its bifullness in $E$ is not sufficient to guarantee the mentioned equality.},
author = {Kalina, Martin},
journal = {Kybernetika},
keywords = {lattice effect algebra; center; atom; MacNeille completion; orthomodular lattice; lattice effect algebra; center; atom; bifullness; state; MacNeille completion},
language = {eng},
number = {6},
pages = {935-947},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras},
url = {http://eudml.org/doc/196610},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Kalina, Martin
TI - Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 935
EP - 947
AB - If element $z$ of a lattice effect algebra $(E,\oplus , {\mathbf {0}}, {\mathbf {1}})$ is central, then the interval $[{\mathbf {0}},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus $. It is a known fact that if the set $C(E)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C(E)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\hat{E}$ of $E$ which is its extension (i.e. $E$ is densely embeddable into $\hat{E}$) then it is possible to embed $E$ into a direct product of irreducible effect algebras. Thus $E$ inherits some of the properties of $\hat{E}$. For example, the existence of a state in $\hat{E}$ implies the existence of a state in $E$. In this context, a natural question arises if the MacNeille completion of the center of $E$ (denoted as ${\mathcal {M}}{\mathcal {C}}(C(E))$) is necessarily the same as the center of $\hat{E}$, i.e., if ${\mathcal {M}}{\mathcal {C}}(C(E))=C(\hat{E})$ is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of $C(E)$ and its bifullness in $E$ is not sufficient to guarantee the mentioned equality.
LA - eng
KW - lattice effect algebra; center; atom; MacNeille completion; orthomodular lattice; lattice effect algebra; center; atom; bifullness; state; MacNeille completion
UR - http://eudml.org/doc/196610
ER -

References

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