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Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras
Kybernetika, Tome 46 (2010) no. 6, pp. 935-947 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 ${\cal M}{\cal C}(C(E))$) is necessarily the same as the center of $\hat{E}$, i.e., if ${\cal M}{\cal C}(C(E))=C(\hat{E})$ 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.
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 ${\cal M}{\cal C}(C(E))$) is necessarily the same as the center of $\hat{E}$, i.e., if ${\cal M}{\cal C}(C(E))=C(\hat{E})$ 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.
Classification : 03G12, 03G27, 06B99
Keywords: lattice effect algebra; center; atom; MacNeille completion 
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     author = {Kalina, Martin},
     title = {Mac {Neille} completion of centers and centers of {Mac} {Neille} completions of lattice effect algebras},
     journal = {Kybernetika},
     pages = {935--947},
     year = {2010},
     volume = {46},
     number = {6},
     mrnumber = {2797418},
     zbl = {1221.06010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_6_a1/}
}
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Kalina, Martin. Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras. Kybernetika, Tome 46 (2010) no. 6, pp. 935-947. http://geodesic.mathdoc.fr/item/KYB_2010_46_6_a1/

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