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Keywords:
lattice effect algebra; center; atom; MacNeille completion
lattice effect algebra; center; atom; MacNeille completion
Summary:
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.
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