Geodesic
On central atoms of Archimedean atomic lattice effect algebras
Kybernetika, Tome 46 (2010) no. 4, pp. 609-620 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 direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C(E)$ is a bifull sublattice of an Archimedean atomic lattice effect algebra $E$. We show that there exists a lattice effect algebra $(E,\oplus, {\mathbf 0}, {\mathbf 1})$ with atomic $C(E)$ which is not a bifull sublattice of $E$. Moreover, we show that also $B(E)$, the center of compatibility, may not be a bifull sublattice of $E$.
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 direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C(E)$ is a bifull sublattice of an Archimedean atomic lattice effect algebra $E$. We show that there exists a lattice effect algebra $(E,\oplus, {\mathbf 0}, {\mathbf 1})$ with atomic $C(E)$ which is not a bifull sublattice of $E$. Moreover, we show that also $B(E)$, the center of compatibility, may not be a bifull sublattice of $E$.
Classification : 03G12, 03G27, 06B99
Keywords: lattice effect algebra; center; atom; bifullness 
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     author = {Kalina, Martin},
     title = {On central atoms of {Archimedean} atomic lattice effect algebras},
     journal = {Kybernetika},
     pages = {609--620},
     year = {2010},
     volume = {46},
     number = {4},
     mrnumber = {2722091},
     zbl = {1214.06002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_4_a2/}
}
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Kalina, Martin. On central atoms of Archimedean atomic lattice effect algebras. Kybernetika, Tome 46 (2010) no. 4, pp. 609-620. http://geodesic.mathdoc.fr/item/KYB_2010_46_4_a2/

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